Abstract group 16128.bb: $C_3\times \SL(2,7).D_8$

• Label: 16128.bb
• Order: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $131$
• Trans deg.: $384$
• Rank: $2$

Group information

Description:	$C_3\times \SL(2,7).D_8$
Order:	\(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	$(C_2\times C_4).C_2^4.\SO(3,7)$, of order \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$, $\PSL(2,7)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56	84	112	168	336
Elements	1	219	170	388	1950	48	928	2456	432	512	96	2528	480	864	2368	192	960	384	384	768	16128
Conjugacy classes	1	4	5	6	14	2	8	18	4	10	4	22	4	8	32	4	8	8	8	16	186
Divisions	1	4	3	6	8	1	6	10	2	3	1	8	2	2	5	1	2	1	1	1	68
Autjugacy classes	1	4	3	5	8	1	7	9	2	3	1	9	2	2	5	1	2	1	1	1	68

Dimension	1	2	3	4	6	7	8	12	14	16	24	28	32	48	56	64	96	128
Irr. complex chars.	12	9	24	0	30	12	36	33	9	21	0	0	0	0	0	0	0	0	186
Irr. rational chars.	4	5	0	2	8	4	5	10	5	5	4	2	2	4	1	3	2	2	68

Minimal presentations

Permutation degree:	$131$
Transitive degree:	$384$
Rank:	$2$
Inequivalent generating pairs:	$1368$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	96
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $131$ $\langle(1,2)(3,12)(4,25)(5,18)(6,38)(7,16)(8,13)(9,45)(10,15)(11,44)(14,51)(17,61) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 2 & 4 & 0 \\ 5 & 1 & 0 & 3 \\ 1 & 0 & 1 & 2 \\ 0 & 6 & 5 & 6 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 2 & 5 & 4 & 2 \\ 3 & 6 & 3 & 4 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 4 & 5 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 6 & 6 \\ 0 & 2 & 5 & 1 \\ 3 & 6 & 5 & 2 \\ 1 & 2 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 4 & 0 \\ 2 & 0 & 1 & 1 \\ 5 & 6 & 2 & 5 \\ 3 & 6 & 6 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 3 & 5 \\ 4 & 6 & 4 & 3 \\ 6 & 0 & 4 & 5 \\ 5 & 6 & 3 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 4 & 6 & 3 \\ 1 & 0 & 1 & 6 \\ 5 & 0 & 3 & 3 \\ 3 & 5 & 6 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_3$ $\, \times\, $ $(\SL(2,7).D_8)$
Semidirect product:	$(D_8.\PSL(2,7))$ $\,\rtimes\,$ $C_6$	$(Q_{16}.\PSL(2,7))$ $\,\rtimes\,$ $C_6$	$(C_{48}.\PSL(2,7))$ $\,\rtimes\,$ $C_2$	$(C_{16}.\PSL(2,7))$ $\,\rtimes\,$ $C_6$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(\SL(2,7):C_6)$ . $D_4$	$(C_3\times \SL(2,7))$ . $D_8$	$\SL(2,7)$ . $(C_3\times D_8)$	$D_8$ . $(C_6\times \GL(3,2))$	all 20

Elements of the group are displayed as matrices in $\GL_{4}(\F_{7})$.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 30208 subgroups in 815 conjugacy classes, 30 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_8\times \GL(3,2)$
Commutator:	$G' \simeq$ $\SL(2,7):C_4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_{24}$	$G/\Phi \simeq$ $C_2^2\times \GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times \SD_{32}$	$G/\operatorname{Fit} \simeq$ $\PSL(2,7)$
Radical:	$R \simeq$ $C_3\times \SD_{32}$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $D_8\times \GL(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{16}:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 16128: $C_3\times \SL(2,7).D_8$

Order 8064: $C_{48}.\PSL(2,7)$, $C_3\times D_8.\PSL(2,7)$, $(C_3\times \SL(2,7)):D_4$

Order 5376: $\SL(2,7).D_8$

Order 4032: $\SL(2,7):C_{12}$, $C_3\times \SL(2,7):C_2^2$, $(C_2\times \SL(2,7)):C_6$

Order 2688: $C_{16}.\PSL(2,7)$, $D_8.\PSL(2,7)$, $Q_{16}.\PSL(2,7)$

Order 2304: $C_3\times \GL(2,3).D_8$ x 2

Order 2016: $\SL(2,7):C_6$ x 2, $C_6\times \SL(2,7)$, $C_{336}:C_6$

Order 1344: $\SL(2,7):C_4$, $\SL(2,7):C_2^2$, $D_4.\PSL(2,7)$

Order 1152: $C_3\times D_8.S_4$ x 2, $C_3\times \GL(2,3):D_4$ x 2, $C_{12}.\GL(2,\mathbb{Z}/4)$ x 2, $C_{12}.\GL(2,\mathbb{Z}/4)$ x 2, $C_{48}.S_4$ x 2, $C_{48}.S_4$ x 2, $C_3\times \SL(2,3).D_8$ x 2

Order 1008: $C_3\times \SL(2,7)$, $C_{168}.C_6$, $C_{168}:C_6$, $C_{21}:C_{48}$

Order 768: $\GL(2,3).D_8$ x 2, $C_{12}.D_4^2$

Order 672: $C_{112}:C_6$ x 3, $\SL(2,7):C_2$ x 2, $C_{21}\times \SD_{32}$, $C_2\times \SL(2,7)$

Order 576: $C_3\times D_4.S_4$ x 2, $C_3\times \GL(2,3):C_2^2$ x 2, $C_3\times D_8.A_4$ x 2, $C_3\times Q_{16}.A_4$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_{24}.S_4$ x 2, $\GL(2,3):C_{12}$ x 2, $C_{24}.S_4$ x 2, $C_{48}.A_4$ x 2, $C_{24}.S_4$ x 2, $C_{48}.D_6$

Order 504: $C_{21}:C_{24}$, $C_{84}.C_6$, $C_{84}:C_6$

Order 384: $\SL(2,3).D_8$ x 4, $D_8.S_4$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_{16}.S_4$ x 2, $C_{16}.S_4$ x 2, $C_{48}.D_4$ x 2, $C_3\times D_4.D_8$ x 2, $C_3\times D_4.D_8$ x 2, $D_8.S_4$ x 2, $\GL(2,3):D_4$ x 2, $C_{48}.C_2^3$ x 2, $C_{48}:D_4$, $C_{24}.D_8$, $C_{24}.D_8$, $D_{16}:C_{12}$, $D_8.C_{24}$, $C_6.D_4^2$, $C_6.D_4^2$

Order 336: $C_{56}.C_6$ x 3, $C_{56}:C_6$ x 3, $C_7:C_{48}$ x 3, $C_{336}$, $\SL(2,7)$, $Q_{16}\times C_{21}$, $D_8\times C_{21}$

Order 288: $\GL(2,3):C_6$ x 4, $\GL(2,3):C_6$ x 4, $Q_8.C_6^2$ x 2, $Q_8.C_6^2$ x 2, $C_{12}.S_4$ x 2, $\GL(2,3):C_6$ x 2, $C_2\times C_6.S_4$ x 2, $C_{24}.A_4$ x 2, $\SL(2,3):C_{12}$ x 2, $D_{24}:C_6$, $C_{24}.D_6$, $C_3^2\times \SD_{32}$, $C_3^2:Q_{32}$, $D_{24}:C_6$, $C_{48}:C_6$, $S_3\times C_{48}$

Order 256: $D_{16}:D_4$

Order 252: $C_{21}:C_{12}$ x 2, $C_{42}:C_6$

Order 224: $C_7\times \SD_{32}$

Order 192: $C_6\times \SD_{32}$ x 4, $C_{16}.A_4$ x 4, $D_8.A_4$ x 4, $Q_{16}.A_4$ x 4, $C_{24}.D_4$ x 3, $C_{24}.D_4$ x 3, $D_8:C_{12}$ x 3, $C_{12}.C_2^4$ x 3, $C_{12}.C_2^4$ x 3, $D_4.S_4$ x 2, $\SL(2,3):D_4$ x 2, $C_8.S_4$ x 2, $\GL(2,3):C_4$ x 2, $C_8.S_4$ x 2, $Q_{32}:C_6$ x 2, $D_{16}:C_6$ x 2, $\OD_{32}:C_6$ x 2, $C_3\times D_4.D_4$ x 2, $C_3\times D_4:D_4$ x 2, $C_8.S_4$ x 2, $C_{12}.D_8$ x 2, $\OD_{32}:C_6$ x 2, $C_{12}.D_8$ x 2, $\OD_{32}:C_6$ x 2, $D_4.S_4$ x 2, $\GL(2,3):C_2^2$ x 2, $D_{16}:C_6$, $C_{12}:Q_{16}$, $C_{12}:D_8$, $\SD_{32}:S_3$, $C_{16}.C_{12}$, $C_8.C_{24}$, $C_4\times C_{48}$

Order 168: $C_{28}.C_6$ x 3, $C_{28}:C_6$ x 3, $C_7:C_{24}$ x 3, $C_{168}$, $Q_8\times C_{21}$, $D_4\times C_{21}$

Order 144: $\SL(2,3):C_6$ x 4, $C_3\times \GL(2,3)$ x 4, $C_6.S_4$ x 4, $C_6\times \SL(2,3)$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_3^2:Q_{16}$, $C_3\times D_{24}$, $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$, $D_{12}:C_6$, $C_{12}.D_6$, $C_3^2\times Q_{16}$, $C_3^2\times D_8$

Order 128: $C_{16}.D_4$ x 2, $D_4.D_8$ x 2, $D_4.D_8$ x 2, $D_{16}:C_2^2$ x 2, $C_{16}:D_4$, $C_8.D_8$, $D_8:D_4$, $D_{16}:C_4$, $C_{16}.D_4$, $C_2.D_4^2$, $D_8:D_4$

Order 126: $C_{21}:C_6$ x 2

Order 112: $C_7\times Q_{16}$, $C_7\times D_8$, $C_{112}$

Order 96: $C_3\times \SD_{32}$ x 8, $Q_{16}:C_6$ x 6, $D_8:C_6$ x 6, $D_8:C_6$ x 5, $C_8.A_4$ x 4, $D_4:C_{12}$ x 4, $D_4.A_4$ x 4, $\SL(2,3):C_2^2$ x 4, $\GL(2,3):C_2$ x 4, $\GL(2,3):C_2$ x 4, $C_6\times Q_{16}$ x 4, $C_6\times D_8$ x 4, $\OD_{16}:C_6$ x 4, $C_8.C_{12}$ x 3, $\Unitary(2,3)$ x 2, $C_3\times \OD_{32}$ x 2, $C_2\times C_{48}$ x 2, $C_{12}.D_4$ x 2, $\OD_{16}:C_6$ x 2, $C_6.C_2^4$ x 2, $C_6.C_2^4$ x 2, $C_4.S_4$ x 2, $\GL(2,3):C_2$ x 2, $C_2^2.S_4$ x 2, $C_{48}:C_2$, $C_3\times Q_{32}$, $C_3\times D_{16}$, $C_4\times C_{24}$, $S_3\times C_{16}$, $C_3:Q_{32}$, $C_3:D_{16}$, $C_{12}:Q_8$, $C_{12}:D_4$, $D_{24}:C_2$, $D_8:S_3$

Order 84: $C_7:C_{12}$ x 6, $C_{14}:C_6$ x 3, $C_{84}$ x 2, $C_2\times C_{42}$

Order 72: $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_3\times \SL(2,3)$ x 2, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3^2:Q_8$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_2\times \SD_{32}$ x 4, $D_4.D_4$ x 3, $D_8:C_2^2$ x 3, $C_8.D_4$ x 3, $C_8.D_4$ x 3, $D_8:C_4$ x 3, $C_4.D_8$ x 2, $\OD_{32}:C_2$ x 2, $C_4.D_8$ x 2, $\OD_{32}:C_2$ x 2, $Q_{32}:C_2$ x 2, $D_{16}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $D_4.D_4$ x 2, $D_4:D_4$ x 2, $C_{16}.C_4$, $C_8.C_8$, $C_4\times C_{16}$, $D_{16}:C_2$, $C_4:Q_{16}$, $C_4:D_8$

Order 63: $C_{21}:C_3$

Order 56: $C_7\times D_4$, $C_{56}$, $C_7\times Q_8$

Order 48: $C_3\times Q_{16}$ x 11, $C_3\times D_8$ x 11, $D_4:C_6$ x 8, $\SL(2,3):C_2$ x 8, $C_3\times \SD_{16}$ x 8, $C_3\times \OD_{16}$ x 6, $C_6\times D_4$ x 5, $C_{48}$ x 5, $C_6\times Q_8$ x 4, $C_2\times \SL(2,3)$ x 4, $\GL(2,3)$ x 4, $C_2.S_4$ x 4, $C_2\times C_{24}$ x 4, $C_3:Q_{16}$, $D_{24}$, $D_{12}:C_2$, $S_3\times C_8$, $D_4:S_3$, $C_4:C_{12}$, $C_4\times C_{12}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:C_{16}$

Order 42: $C_7:C_6$ x 6, $C_{42}$ x 2

Order 36: $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 32: $Q_{16}:C_2$ x 6, $D_8:C_2$ x 6, $\SD_{32}$ x 6, $D_8:C_2$ x 5, $C_2\times Q_{16}$ x 4, $C_2\times D_8$ x 4, $\OD_{16}:C_2$ x 4, $C_4\wr C_2$ x 4, $C_8.C_4$ x 3, $C_4.D_4$ x 2, $\OD_{16}:C_2$ x 2, $C_4.C_2^3$ x 2, $D_4:C_2^2$ x 2, $\OD_{32}$ x 2, $C_2\times C_{16}$ x 2, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $Q_{32}$, $D_{16}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_3\times D_4$ x 12, $C_{24}$ x 8, $C_3\times Q_8$ x 8, $C_2\times C_{12}$ x 6, $\SL(2,3)$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2, $C_3:D_4$, $C_6:C_4$, $C_3:Q_8$, $C_3:C_8$

Order 21: $C_7:C_3$ x 3, $C_{21}$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $Q_{16}$ x 9, $D_8$ x 9, $\SD_{16}$ x 8, $D_4:C_2$ x 8, $\OD_{16}$ x 6, $C_2\times D_4$ x 5, $C_2\times C_8$ x 4, $C_2\times Q_8$ x 4, $C_{16}$ x 3, $C_4:C_4$, $C_4^2$

Order 14: $C_{14}$ x 2

Order 12: $C_{12}$ x 10, $C_2\times C_6$ x 7, $D_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 10, $Q_8$ x 6, $C_2\times C_4$ x 6, $C_8$ x 6, $C_2^3$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 8, $S_3$ x 2

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 16128: $C_3\times \SL(2,7).D_8$

Order 8064: $C_{48}.\PSL(2,7)$, $C_3\times D_8.\PSL(2,7)$, $(C_3\times \SL(2,7)):D_4$

Order 5376: $\SL(2,7).D_8$

Order 4032: $\SL(2,7):C_{12}$, $C_3\times \SL(2,7):C_2^2$, $(C_2\times \SL(2,7)):C_6$

Order 2688: $C_{16}.\PSL(2,7)$, $D_8.\PSL(2,7)$, $Q_{16}.\PSL(2,7)$

Order 2304: $C_3\times \GL(2,3).D_8$

Order 2016: $\SL(2,7):C_6$ x 2, $C_6\times \SL(2,7)$, $C_{336}:C_6$

Order 1344: $\SL(2,7):C_4$, $\SL(2,7):C_2^2$, $D_4.\PSL(2,7)$

Order 1152: $C_3\times D_8.S_4$, $C_3\times \GL(2,3):D_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{48}.S_4$, $C_{48}.S_4$, $C_3\times \SL(2,3).D_8$

Order 1008: $C_3\times \SL(2,7)$, $C_{168}.C_6$, $C_{168}:C_6$, $C_{21}:C_{48}$

Order 768: $C_{12}.D_4^2$, $\GL(2,3).D_8$

Order 672: $C_{112}:C_6$ x 2, $\SL(2,7):C_2$ x 2, $C_{21}\times \SD_{32}$, $C_2\times \SL(2,7)$

Order 576: $C_3\times D_4.S_4$, $C_3\times \GL(2,3):C_2^2$, $C_3\times D_8.A_4$, $C_3\times Q_{16}.A_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}.S_4$, $\GL(2,3):C_{12}$, $C_{24}.S_4$, $C_{48}.D_6$, $C_{48}.A_4$, $C_{24}.S_4$

Order 504: $C_{21}:C_{24}$, $C_{84}.C_6$, $C_{84}:C_6$

Order 384: $\SL(2,3).D_8$ x 2, $D_8.S_4$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_{16}.S_4$, $C_{16}.S_4$, $C_{48}:D_4$, $C_{48}.D_4$, $C_{24}.D_8$, $C_{24}.D_8$, $C_3\times D_4.D_8$, $C_3\times D_4.D_8$, $D_{16}:C_{12}$, $D_8.C_{24}$, $D_8.S_4$, $\GL(2,3):D_4$, $C_{48}.C_2^3$, $C_6.D_4^2$, $C_6.D_4^2$

Order 336: $C_{56}.C_6$ x 2, $C_{56}:C_6$ x 2, $C_7:C_{48}$ x 2, $C_{336}$, $\SL(2,7)$, $Q_{16}\times C_{21}$, $D_8\times C_{21}$

Order 288: $\GL(2,3):C_6$ x 2, $\GL(2,3):C_6$ x 2, $Q_8.C_6^2$, $Q_8.C_6^2$, $C_{12}.S_4$, $\GL(2,3):C_6$, $C_2\times C_6.S_4$, $D_{24}:C_6$, $C_{24}.D_6$, $C_{24}.A_4$, $\SL(2,3):C_{12}$, $C_3^2\times \SD_{32}$, $C_3^2:Q_{32}$, $D_{24}:C_6$, $C_{48}:C_6$, $S_3\times C_{48}$

Order 256: $D_{16}:D_4$

Order 252: $C_{21}:C_{12}$ x 2, $C_{42}:C_6$

Order 224: $C_7\times \SD_{32}$

Order 192: $D_8:C_{12}$ x 3, $C_6\times \SD_{32}$ x 2, $C_{24}.D_4$ x 2, $C_{24}.D_4$ x 2, $C_{16}.A_4$ x 2, $C_{12}.D_8$ x 2, $C_{12}.C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $D_8.A_4$ x 2, $Q_{16}.A_4$ x 2, $D_4.S_4$, $\SL(2,3):D_4$, $C_8.S_4$, $\GL(2,3):C_4$, $C_8.S_4$, $Q_{32}:C_6$, $D_{16}:C_6$, $D_{16}:C_6$, $\OD_{32}:C_6$, $C_{12}:Q_{16}$, $C_{12}:D_8$, $C_3\times D_4.D_4$, $C_3\times D_4:D_4$, $\SD_{32}:S_3$, $C_8.S_4$, $C_{16}.C_{12}$, $C_8.C_{24}$, $C_{12}.D_8$, $\OD_{32}:C_6$, $\OD_{32}:C_6$, $C_4\times C_{48}$, $D_4.S_4$, $\GL(2,3):C_2^2$

Order 168: $C_{28}.C_6$ x 2, $C_{28}:C_6$ x 2, $C_7:C_{24}$ x 2, $C_{168}$, $Q_8\times C_{21}$, $D_4\times C_{21}$

Order 144: $\SL(2,3):C_6$ x 2, $C_3\times \GL(2,3)$ x 2, $C_6.S_4$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_3^2:Q_{16}$, $C_3\times D_{24}$, $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$, $D_{12}:C_6$, $C_{12}.D_6$, $C_6\times \SL(2,3)$, $C_3^2\times Q_{16}$, $C_3^2\times D_8$

Order 128: $C_{16}:D_4$, $C_{16}.D_4$, $C_8.D_8$, $D_8:D_4$, $D_4.D_8$, $D_4.D_8$, $D_{16}:C_4$, $C_{16}.D_4$, $D_{16}:C_2^2$, $C_2.D_4^2$, $D_8:D_4$

Order 126: $C_{21}:C_6$ x 2

Order 112: $C_7\times Q_{16}$, $C_7\times D_8$, $C_{112}$

Order 96: $C_3\times \SD_{32}$ x 6, $D_8:C_6$ x 5, $C_8.C_{12}$ x 3, $D_4:C_{12}$ x 3, $Q_{16}:C_6$ x 3, $D_8:C_6$ x 3, $C_6\times Q_{16}$ x 3, $C_6\times D_8$ x 3, $\OD_{16}:C_6$ x 3, $C_8.A_4$ x 2, $C_2\times C_{48}$ x 2, $D_4.A_4$ x 2, $\SL(2,3):C_2^2$ x 2, $\GL(2,3):C_2$ x 2, $\GL(2,3):C_2$ x 2, $C_{48}:C_2$, $\Unitary(2,3)$, $C_3\times Q_{32}$, $C_3\times D_{16}$, $C_3\times \OD_{32}$, $C_{12}.D_4$, $\OD_{16}:C_6$, $C_4\times C_{24}$, $S_3\times C_{16}$, $C_3:Q_{32}$, $C_3:D_{16}$, $C_6.C_2^4$, $C_6.C_2^4$, $C_4.S_4$, $\GL(2,3):C_2$, $C_2^2.S_4$, $C_{12}:Q_8$, $C_{12}:D_4$, $D_{24}:C_2$, $D_8:S_3$

Order 84: $C_7:C_{12}$ x 4, $C_{14}:C_6$ x 2, $C_{84}$ x 2, $C_2\times C_{42}$

Order 72: $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $D_8:C_4$ x 3, $C_4.D_8$ x 2, $D_4.D_4$ x 2, $D_8:C_2^2$ x 2, $C_2\times \SD_{32}$ x 2, $C_8.D_4$ x 2, $C_8.D_4$ x 2, $C_{16}.C_4$, $C_8.C_8$, $C_4.D_8$, $\OD_{32}:C_2$, $\OD_{32}:C_2$, $C_4\times C_{16}$, $Q_{32}:C_2$, $D_{16}:C_2$, $D_{16}:C_2$, $\OD_{32}:C_2$, $C_4:Q_{16}$, $C_4:D_8$, $D_4.D_4$, $D_4:D_4$

Order 63: $C_{21}:C_3$

Order 56: $C_7\times D_4$, $C_{56}$, $C_7\times Q_8$

Order 48: $C_3\times Q_{16}$ x 9, $C_3\times D_8$ x 9, $D_4:C_6$ x 5, $C_3\times \SD_{16}$ x 5, $C_3\times \OD_{16}$ x 5, $C_{48}$ x 5, $\SL(2,3):C_2$ x 4, $C_2\times C_{24}$ x 4, $C_6\times D_4$ x 3, $C_6\times Q_8$ x 2, $C_2\times \SL(2,3)$ x 2, $\GL(2,3)$ x 2, $C_2.S_4$ x 2, $C_3:Q_{16}$, $D_{24}$, $D_{12}:C_2$, $S_3\times C_8$, $D_4:S_3$, $C_4:C_{12}$, $C_4\times C_{12}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:C_{16}$

Order 42: $C_7:C_6$ x 4, $C_{42}$ x 2

Order 36: $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 32: $D_8:C_2$ x 5, $\SD_{32}$ x 4, $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times Q_{16}$ x 3, $C_2\times D_8$ x 3, $\OD_{16}:C_2$ x 3, $C_8.C_4$ x 3, $C_4\wr C_2$ x 3, $C_2\times C_{16}$ x 2, $C_4.D_4$, $\OD_{16}:C_2$, $C_4.C_2^3$, $D_4:C_2^2$, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $Q_{32}$, $D_{16}$, $\OD_{32}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_3\times D_4$ x 9, $C_{24}$ x 8, $C_3\times Q_8$ x 6, $C_2\times C_{12}$ x 5, $D_{12}$ x 2, $C_4\times S_3$ x 2, $\SL(2,3)$ x 2, $C_3:D_4$, $C_6:C_4$, $C_3:Q_8$, $C_2^2\times C_6$, $C_3:C_8$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $Q_{16}$ x 7, $D_8$ x 7, $\SD_{16}$ x 5, $\OD_{16}$ x 5, $D_4:C_2$ x 5, $C_2\times C_8$ x 4, $C_2\times D_4$ x 3, $C_{16}$ x 3, $C_2\times Q_8$ x 2, $C_4:C_4$, $C_4^2$

Order 14: $C_{14}$ x 2

Order 12: $C_{12}$ x 9, $C_2\times C_6$ x 6, $D_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 7, $C_8$ x 6, $C_2\times C_4$ x 5, $Q_8$ x 4, $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 8, $S_3$ x 2

Order 4: $C_4$ x 5, $C_2^2$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16128: $C_3\times \SL(2,7).D_8$ ($C_1$)

Order 8064: $C_{48}.\PSL(2,7)$ ($C_2$), $C_3\times D_8.\PSL(2,7)$ ($C_2$), $(C_3\times \SL(2,7)):D_4$ ($C_2$)

Order 5376: $\SL(2,7).D_8$ ($C_3$)

Order 4032: $\SL(2,7):C_{12}$ ($C_2^2$)

Order 2688: $C_{16}.\PSL(2,7)$ ($C_6$), $D_8.\PSL(2,7)$ ($C_6$), $Q_{16}.\PSL(2,7)$ ($C_6$)

Order 2016: $\SL(2,7):C_6$ ($D_4$)

Order 1344: $\SL(2,7):C_4$ ($C_2\times C_6$)

Order 1008: $C_3\times \SL(2,7)$ ($D_8$)

Order 672: $\SL(2,7):C_2$ ($C_3\times D_4$)

Order 336: $\SL(2,7)$ ($C_3\times D_8$)

Order 96: $C_3\times \SD_{32}$ ($\PSL(2,7)$)

Order 48: $C_3\times Q_{16}$ ($C_2\times \GL(3,2)$), $C_3\times D_8$ ($C_2\times \GL(3,2)$), $C_{48}$ ($C_2\times \GL(3,2)$)

Order 32: $\SD_{32}$ ($C_3\times \GL(3,2)$)

Order 24: $C_{24}$ ($C_2^2\times \GL(3,2)$)

Order 16: $Q_{16}$ ($C_6\times \GL(3,2)$), $D_8$ ($C_6\times \GL(3,2)$), $C_{16}$ ($C_6\times \GL(3,2)$)

Order 12: $C_{12}$ ($D_4\times \GL(3,2)$)

Order 8: $C_8$ ($C_2\times C_6\times \GL(3,2)$)

Order 6: $C_6$ ($D_8\times \GL(3,2)$)

Order 4: $C_4$ ($C_3\times D_4\times \GL(3,2)$)

Order 3: $C_3$ ($\SL(2,7).D_8$)

Order 2: $C_2$ ($(C_3\times D_8).\PSL(2,7)$)

Order 1: $C_1$ ($C_3\times \SL(2,7).D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16128: $C_3\times \SL(2,7).D_8$ ($C_1$)

Order 8064: $C_{48}.\PSL(2,7)$ ($C_2$), $C_3\times D_8.\PSL(2,7)$ ($C_2$), $(C_3\times \SL(2,7)):D_4$ ($C_2$)

Order 5376: $\SL(2,7).D_8$ ($C_3$)

Order 4032: $\SL(2,7):C_{12}$ ($C_2^2$)

Order 2688: $C_{16}.\PSL(2,7)$ ($C_6$), $D_8.\PSL(2,7)$ ($C_6$), $Q_{16}.\PSL(2,7)$ ($C_6$)

Order 2016: $\SL(2,7):C_6$ ($D_4$)

Order 1344: $\SL(2,7):C_4$ ($C_2\times C_6$)

Order 1008: $C_3\times \SL(2,7)$ ($D_8$)

Order 672: $\SL(2,7):C_2$ ($C_3\times D_4$)

Order 336: $\SL(2,7)$ ($C_3\times D_8$)

Order 96: $C_3\times \SD_{32}$ ($\PSL(2,7)$)

Order 48: $C_3\times Q_{16}$ ($C_2\times \GL(3,2)$), $C_3\times D_8$ ($C_2\times \GL(3,2)$), $C_{48}$ ($C_2\times \GL(3,2)$)

Order 32: $\SD_{32}$ ($C_3\times \GL(3,2)$)

Order 24: $C_{24}$ ($C_2^2\times \GL(3,2)$)

Order 16: $Q_{16}$ ($C_6\times \GL(3,2)$), $D_8$ ($C_6\times \GL(3,2)$), $C_{16}$ ($C_6\times \GL(3,2)$)

Order 12: $C_{12}$ ($D_4\times \GL(3,2)$)

Order 8: $C_8$ ($C_2\times C_6\times \GL(3,2)$)

Order 6: $C_6$ ($D_8\times \GL(3,2)$)

Order 4: $C_4$ ($C_3\times D_4\times \GL(3,2)$)

Order 3: $C_3$ ($\SL(2,7).D_8$)

Order 2: $C_2$ ($(C_3\times D_8).\PSL(2,7)$)

Order 1: $C_1$ ($C_3\times \SL(2,7).D_8$)

Series

Derived series

$C_3\times \SL(2,7).D_8$

$\rhd$

$\SL(2,7):C_4$

$\rhd$

$\SL(2,7)$

Chief series

$C_3\times \SL(2,7).D_8$

$\rhd$

$C_3\times \SD_{32}$

$\rhd$

$C_3\times D_8$

$\rhd$

$C_{24}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_3\times \SL(2,7).D_8$

$\rhd$

$\SL(2,7):C_4$

$\rhd$

$\SL(2,7):C_2$

$\rhd$

$\SL(2,7)$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_{24}$

$\lhd$

$C_3\times \SD_{32}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $186 \times 186$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.