Abstract group 1344.9877: $C_2^2\times C_6.D_{28}$

• Label: 1344.9877
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \cdot 3^{2} \)
• Perm deg.: $22$
• Trans deg.: $672$
• Rank: $4$

Group information

Description:	$C_2^2\times C_6.D_{28}$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^8.C_2^4.C_7.C_3^3.C_2^3$, of order \(6193152\)\(\medspace = 2^{15} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	7	14	21	28	42
Elements	1	127	2	384	254	6	90	12	288	180	1344
Conjugacy classes	1	23	1	16	31	3	45	3	48	45	216
Divisions	1	23	1	8	23	1	15	1	4	15	92
Autjugacy classes	1	5	1	2	5	1	4	1	1	4	25

Dimension	1	2	4	6	12
Irr. complex chars.	32	136	48	0	0	216
Irr. rational chars.	16	32	8	8	28	92

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$672$
Rank:	$4$
Inequivalent generating quadruples:	$155610$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{4}=d^{6}=e^{14}=[a,b]=[a,c]=[a,d]=[a,e]=[b,c]=[b,d]=[b,e]=[c,e]=1, d^{c}=d^{5}e^{7}, e^{d}=e^{13} \rangle$
Permutation group:	Degree $22$ $\langle(2,3)(4,5)(6,7)(20,22), (8,9)(10,11)(13,14)(15,16,17,18)(19,20,21,22), (8,10) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 31 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 42 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 43 & 49 \\ 42 & 29 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/84\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_6.D_{28})$
Semidirect product:	$(C_6.C_2^4)$ $\,\rtimes\,$ $D_7$	$C_7$ $\,\rtimes\,$ $(C_2^4.D_6)$	$(C_2^3.D_{42})$ $\,\rtimes\,$ $C_2$	$(C_6.C_2^3)$ $\,\rtimes\,$ $D_{14}$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(S_3\times D_7)$	$(C_2^3\times D_{14})$ . $S_3$	$C_2^3$ . $(S_3\times D_{14})$	$(C_2^2\times D_{14})$ . $D_6$	all 48

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{84}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 8148 subgroups in 1348 conjugacy classes, 432 normal (30 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $S_3\times D_7$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_2^2\times C_6.D_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1344: $C_2^2\times C_6.D_{28}$

Order 672: $C_2\times C_6.D_{28}$ x 12, $C_{21}:C_2^5$, $C_2^3.D_{42}$, $C_{42}.C_2^4$

Order 448: $C_2^3.D_{28}$

Order 336: $C_{42}:C_2^3$ x 22, $C_6.D_{28}$ x 16, $C_2^2.D_{42}$ x 10, $C_{42}.C_2^3$ x 10, $C_2^3\times C_{42}$

Order 224: $C_2^2.D_{28}$ x 12, $C_2^3\times D_{14}$, $C_2^3\times C_{28}$, $C_{14}.C_2^4$

Order 192: $C_2^4.D_6$

Order 168: $C_6\times D_{14}$ x 84, $C_{42}:C_4$ x 16, $C_6:C_{28}$ x 16, $C_2^2\times C_{42}$ x 15

Order 112: $C_2^2\times D_{14}$ x 22, $D_{14}:C_4$ x 16, $C_2^2\times C_{28}$ x 10, $C_2^2.D_{14}$ x 10, $C_2^3\times C_{14}$

Order 96: $C_2^3.D_6$ x 12, $C_6.C_2^4$ x 2, $C_2^4\times C_6$

Order 84: $C_3\times D_{14}$ x 64, $C_2\times C_{42}$ x 35, $C_{21}:C_4$ x 4, $C_3:C_{28}$ x 4

Order 64: $C_2^4:C_4$

Order 56: $C_2\times D_{14}$ x 84, $C_2\times C_{28}$ x 16, $C_{14}:C_4$ x 16, $C_2^2\times C_{14}$ x 15

Order 48: $C_2^3\times C_6$ x 23, $C_6.C_2^3$ x 20, $C_6.D_4$ x 16

Order 42: $C_{42}$ x 15, $C_3\times D_7$ x 8

Order 32: $C_2^3:C_4$ x 12, $C_2^3\times C_4$ x 2, $C_2^5$

Order 28: $D_{14}$ x 64, $C_2\times C_{14}$ x 35, $C_{28}$ x 4, $C_7:C_4$ x 4

Order 24: $C_2^2\times C_6$ x 99, $C_6:C_4$ x 32

Order 21: $C_{21}$

Order 16: $C_2^4$ x 23, $C_2^2\times C_4$ x 20, $C_2^2:C_4$ x 16

Order 14: $C_{14}$ x 15, $D_7$ x 8

Order 12: $C_2\times C_6$ x 99, $C_3:C_4$ x 8

Order 8: $C_2^3$ x 99, $C_2\times C_4$ x 32

Order 7: $C_7$

Order 6: $C_6$ x 23

Order 4: $C_2^2$ x 99, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_2^2\times C_6.D_{28}$

Order 672: $C_2\times C_6.D_{28}$, $C_{21}:C_2^5$, $C_2^3.D_{42}$, $C_{42}.C_2^4$

Order 448: $C_2^3.D_{28}$

Order 336: $C_{42}:C_2^3$ x 4, $C_2^2.D_{42}$ x 2, $C_{42}.C_2^3$ x 2, $C_6.D_{28}$, $C_2^3\times C_{42}$

Order 224: $C_2^3\times D_{14}$, $C_2^3\times C_{28}$, $C_{14}.C_2^4$, $C_2^2.D_{28}$

Order 192: $C_2^4.D_6$

Order 168: $C_6\times D_{14}$ x 5, $C_2^2\times C_{42}$ x 4, $C_{42}:C_4$ x 2, $C_6:C_{28}$ x 2

Order 112: $C_2^2\times D_{14}$ x 4, $C_2^2\times C_{28}$ x 2, $C_2^2.D_{14}$ x 2, $C_2^3\times C_{14}$, $D_{14}:C_4$

Order 96: $C_6.C_2^4$ x 2, $C_2^4\times C_6$, $C_2^3.D_6$

Order 84: $C_2\times C_{42}$ x 5, $C_3\times D_{14}$ x 4, $C_{21}:C_4$, $C_3:C_{28}$

Order 64: $C_2^4:C_4$

Order 56: $C_2\times D_{14}$ x 5, $C_2^2\times C_{14}$ x 4, $C_2\times C_{28}$ x 2, $C_{14}:C_4$ x 2

Order 48: $C_2^3\times C_6$ x 5, $C_6.C_2^3$ x 4, $C_6.D_4$

Order 42: $C_{42}$ x 4, $C_3\times D_7$

Order 32: $C_2^3\times C_4$ x 2, $C_2^5$, $C_2^3:C_4$

Order 28: $C_2\times C_{14}$ x 5, $D_{14}$ x 4, $C_{28}$, $C_7:C_4$

Order 24: $C_2^2\times C_6$ x 9, $C_6:C_4$ x 4

Order 21: $C_{21}$

Order 16: $C_2^4$ x 5, $C_2^2\times C_4$ x 4, $C_2^2:C_4$

Order 14: $C_{14}$ x 4, $D_7$

Order 12: $C_2\times C_6$ x 9, $C_3:C_4$ x 2

Order 8: $C_2^3$ x 9, $C_2\times C_4$ x 4

Order 7: $C_7$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_2^2\times C_6.D_{28}$ ($C_1$)

Order 672: $C_2\times C_6.D_{28}$ ($C_2$) x 12, $C_{21}:C_2^5$ ($C_2$), $C_2^3.D_{42}$ ($C_2$), $C_{42}.C_2^4$ ($C_2$)

Order 336: $C_6.D_{28}$ ($C_2^2$) x 16, $C_{42}:C_2^3$ ($C_4$) x 8, $C_{42}:C_2^3$ ($C_2^2$) x 6, $C_2^2.D_{42}$ ($C_2^2$) x 6, $C_{42}.C_2^3$ ($C_2^2$) x 6, $C_2^3\times C_{42}$ ($C_2^2$)

Order 224: $C_2^3\times D_{14}$ ($S_3$)

Order 168: $C_6\times D_{14}$ ($C_2\times C_4$) x 24, $C_2^2\times C_{42}$ ($D_4$) x 8, $C_2^2\times C_{42}$ ($C_2\times C_4$) x 4, $C_6\times D_{14}$ ($C_2^3$) x 4, $C_{42}:C_4$ ($C_2^3$) x 4, $C_6:C_{28}$ ($C_2^3$) x 4, $C_2^2\times C_{42}$ ($C_2^3$) x 3

Order 112: $C_2^2\times D_{14}$ ($C_3:C_4$) x 8, $C_2^2\times D_{14}$ ($D_6$) x 6, $C_2^3\times C_{14}$ ($D_6$)

Order 96: $C_6.C_2^4$ ($D_7$)

Order 84: $C_2\times C_{42}$ ($C_2^2:C_4$) x 16, $C_2\times C_{42}$ ($C_2\times D_4$) x 12, $C_3\times D_{14}$ ($C_2^2\times C_4$) x 8, $C_2\times C_{42}$ ($C_2^2\times C_4$) x 6, $C_2\times C_{42}$ ($C_2^4$)

Order 56: $C_2\times D_{14}$ ($C_6:C_4$) x 24, $C_2^2\times C_{14}$ ($C_3:D_4$) x 8, $C_2^2\times C_{14}$ ($C_6:C_4$) x 4, $C_2\times D_{14}$ ($C_2\times D_6$) x 4, $C_2^2\times C_{14}$ ($C_2\times D_6$) x 3

Order 48: $C_6.C_2^3$ ($D_{14}$) x 6, $C_2^3\times C_6$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^3:C_4$) x 12, $C_{42}$ ($C_2^2\times D_4$) x 2, $C_{42}$ ($C_2^3\times C_4$)

Order 28: $C_2\times C_{14}$ ($C_6.D_4$) x 16, $C_2\times C_{14}$ ($C_6:D_4$) x 12, $D_{14}$ ($C_6.C_2^3$) x 8, $C_2\times C_{14}$ ($C_6.C_2^3$) x 6, $C_2\times C_{14}$ ($C_2^2\times D_6$)

Order 24: $C_6:C_4$ ($C_2\times D_{14}$) x 4, $C_2^2\times C_6$ ($C_7:D_4$) x 4, $C_2^2\times C_6$ ($D_{28}$) x 4, $C_2^2\times C_6$ ($C_4\times D_7$) x 4, $C_2^2\times C_6$ ($C_2\times D_{14}$) x 3

Order 21: $C_{21}$ ($C_2^4:C_4$)

Order 16: $C_2^4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_2^3.D_6$) x 12, $C_{14}$ ($C_2^3:D_6$) x 2, $C_{14}$ ($C_6.C_2^4$)

Order 12: $C_2\times C_6$ ($D_{14}:C_4$) x 16, $C_2\times C_6$ ($C_{14}:D_4$) x 6, $C_2\times C_6$ ($C_2\times D_{28}$) x 6, $C_2\times C_6$ ($C_4\times D_{14}$) x 6, $C_2\times C_6$ ($C_2^2\times D_{14}$)

Order 8: $C_2^3$ ($C_3:D_{28}$) x 4, $C_2^3$ ($C_{21}:D_4$) x 4, $C_2^3$ ($C_6.D_{14}$) x 4, $C_2^3$ ($S_3\times D_{14}$) x 3

Order 7: $C_7$ ($C_2^4.D_6$)

Order 6: $C_6$ ($C_2^2.D_{28}$) x 12, $C_6$ ($C_2^3:D_{14}$), $C_6$ ($C_2^2\times D_{28}$), $C_6$ ($C_{28}:C_2^3$)

Order 4: $C_2^2$ ($C_6.D_{28}$) x 16, $C_2^2$ ($C_6:D_{28}$) x 6, $C_2^2$ ($C_{42}:D_4$) x 6, $C_2^2$ ($D_{14}.D_6$) x 6, $C_2^2$ ($D_6\times D_{14}$)

Order 3: $C_3$ ($C_2^3.D_{28}$)

Order 2: $C_2$ ($C_2\times C_6.D_{28}$) x 12, $C_2$ ($D_{42}:C_2^3$), $C_2$ ($C_{42}.C_2^4$), $C_2$ ($C_{42}.C_2^4$)

Order 1: $C_1$ ($C_2^2\times C_6.D_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_2^2\times C_6.D_{28}$ ($C_1$)

Order 672: $C_2\times C_6.D_{28}$ ($C_2$), $C_{21}:C_2^5$ ($C_2$), $C_2^3.D_{42}$ ($C_2$), $C_{42}.C_2^4$ ($C_2$)

Order 336: $C_6.D_{28}$ ($C_2^2$), $C_2^3\times C_{42}$ ($C_2^2$), $C_{42}:C_2^3$ ($C_2^2$), $C_{42}:C_2^3$ ($C_4$), $C_2^2.D_{42}$ ($C_2^2$), $C_{42}.C_2^3$ ($C_2^2$)

Order 224: $C_2^3\times D_{14}$ ($S_3$)

Order 168: $C_2^2\times C_{42}$ ($D_4$) x 2, $C_2^2\times C_{42}$ ($C_2^3$), $C_2^2\times C_{42}$ ($C_2\times C_4$), $C_6\times D_{14}$ ($C_2^3$), $C_6\times D_{14}$ ($C_2\times C_4$), $C_{42}:C_4$ ($C_2^3$), $C_6:C_{28}$ ($C_2^3$)

Order 112: $C_2^3\times C_{14}$ ($D_6$), $C_2^2\times D_{14}$ ($D_6$), $C_2^2\times D_{14}$ ($C_3:C_4$)

Order 96: $C_6.C_2^4$ ($D_7$)

Order 84: $C_2\times C_{42}$ ($C_2\times D_4$) x 2, $C_2\times C_{42}$ ($C_2^2:C_4$), $C_2\times C_{42}$ ($C_2^4$), $C_2\times C_{42}$ ($C_2^2\times C_4$), $C_3\times D_{14}$ ($C_2^2\times C_4$)

Order 56: $C_2^2\times C_{14}$ ($C_3:D_4$) x 2, $C_2^2\times C_{14}$ ($C_6:C_4$), $C_2^2\times C_{14}$ ($C_2\times D_6$), $C_2\times D_{14}$ ($C_6:C_4$), $C_2\times D_{14}$ ($C_2\times D_6$)

Order 48: $C_2^3\times C_6$ ($D_{14}$), $C_6.C_2^3$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^2\times D_4$) x 2, $C_{42}$ ($C_2^3\times C_4$), $C_{42}$ ($C_2^3:C_4$)

Order 28: $C_2\times C_{14}$ ($C_6:D_4$) x 2, $C_2\times C_{14}$ ($C_2^2\times D_6$), $C_2\times C_{14}$ ($C_6.C_2^3$), $C_2\times C_{14}$ ($C_6.D_4$), $D_{14}$ ($C_6.C_2^3$)

Order 24: $C_6:C_4$ ($C_2\times D_{14}$), $C_2^2\times C_6$ ($C_7:D_4$), $C_2^2\times C_6$ ($D_{28}$), $C_2^2\times C_6$ ($C_4\times D_7$), $C_2^2\times C_6$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($C_2^4:C_4$)

Order 16: $C_2^4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_2^3:D_6$) x 2, $C_{14}$ ($C_6.C_2^4$), $C_{14}$ ($C_2^3.D_6$)

Order 12: $C_2\times C_6$ ($C_2^2\times D_{14}$), $C_2\times C_6$ ($C_{14}:D_4$), $C_2\times C_6$ ($C_2\times D_{28}$), $C_2\times C_6$ ($C_4\times D_{14}$), $C_2\times C_6$ ($D_{14}:C_4$)

Order 8: $C_2^3$ ($S_3\times D_{14}$), $C_2^3$ ($C_3:D_{28}$), $C_2^3$ ($C_{21}:D_4$), $C_2^3$ ($C_6.D_{14}$)

Order 7: $C_7$ ($C_2^4.D_6$)

Order 6: $C_6$ ($C_2^3:D_{14}$), $C_6$ ($C_2^2\times D_{28}$), $C_6$ ($C_{28}:C_2^3$), $C_6$ ($C_2^2.D_{28}$)

Order 4: $C_2^2$ ($C_6.D_{28}$), $C_2^2$ ($D_6\times D_{14}$), $C_2^2$ ($C_6:D_{28}$), $C_2^2$ ($C_{42}:D_4$), $C_2^2$ ($D_{14}.D_6$)

Order 3: $C_3$ ($C_2^3.D_{28}$)

Order 2: $C_2$ ($C_2\times C_6.D_{28}$), $C_2$ ($D_{42}:C_2^3$), $C_2$ ($C_{42}.C_2^4$), $C_2$ ($C_{42}.C_2^4$)

Order 1: $C_1$ ($C_2^2\times C_6.D_{28}$)

Series

Derived series

$C_2^2\times C_6.D_{28}$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_2^2\times C_6.D_{28}$

$\rhd$

$C_{42}.C_2^4$

$\rhd$

$C_2^3\times C_{42}$

$\rhd$

$C_2^2\times C_{42}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2^2\times C_6.D_{28}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^4$

Character theory

Complex character table

See the $216 \times 216$ 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 $92 \times 92$ rational character table.