Abstract group 384.17366: $C_2\times D_4\times C_{24}$

• Label: 384.17366
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \)
• Perm deg.: $17$
• Trans deg.: $192$
• Rank: $4$

Group information

Description:	$C_2\times D_4\times C_{24}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^8.C_2^6$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 7, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	23	2	40	46	64	80	128	384
Conjugacy classes	1	15	2	24	30	40	48	80	240
Divisions	1	15	1	14	15	10	14	10	80
Autjugacy classes	1	5	1	6	5	3	6	3	30

Dimension	1	2	4	8	16
Irr. complex chars.	192	48	0	0	0	240
Irr. rational chars.	16	28	22	12	2	80

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$192$
Rank:	$4$
Inequivalent generating quadruples:	$403200$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{8}=d^{12}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, d^{b}=d^{7} \rangle$
Permutation group:	Degree $17$ $\langle(2,4)(7,8,10,12,9,11,13,14), (1,2)(3,4)(5,6)(7,9)(8,11)(10,13)(12,14), (1,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 5 \\ 10 & 13 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 12 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 19 & 10 \\ 10 & 9 \end{array}\right), \left(\begin{array}{rr} 13 & 16 \\ 12 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_8$ $\, \times\, $ $C_3$ $\, \times\, $ $D_4$
Semidirect product:	$(C_2^3\times C_8)$ $\,\rtimes\,$ $C_6$	$(C_2^3:C_8)$ $\,\rtimes\,$ $C_6$	$(C_4^2.C_4)$ $\,\rtimes\,$ $C_6$	$(C_2^3\times C_{24})$ $\,\rtimes\,$ $C_2$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4\times C_{12})$ . $C_4$	$(C_4\times D_4)$ . $C_{12}$	$C_{12}$ . $(C_4\times D_4)$	$C_4$ . $(D_4\times C_{12})$	all 67

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{3} \times C_{24} \simeq C_{2}^{3} \times C_{8} \times C_{3}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 840 subgroups in 616 conjugacy classes, 392 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{24}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3\times C_{24}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times D_4\times C_{24}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times D_4\times C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 384: $C_2\times D_4\times C_{24}$

Order 192: $D_4\times C_{24}$ x 8, $C_2^3:C_{24}$ x 2, $C_2^3\times C_{24}$ x 2, $C_4^2.C_{12}$, $C_2\times C_4\times C_{24}$, $C_2\times D_4\times C_{12}$

Order 128: $C_4^2.C_2^3$

Order 96: $C_2^2\times C_{24}$ x 20, $C_2^2:C_{24}$ x 8, $D_4\times C_{12}$ x 8, $C_4:C_{24}$ x 4, $C_4\times C_{24}$ x 4, $C_2^3\times C_{12}$ x 2, $C_2^3:C_{12}$ x 2, $C_{12}:C_2^3$, $C_2\times C_4:C_{12}$, $C_2\times C_4\times C_{12}$

Order 64: $C_8\times D_4$ x 8, $C_2^3:C_8$ x 2, $C_2^3\times C_8$ x 2, $C_2\times C_4\times C_8$, $C_4^2:C_2^2$, $C_4^2.C_4$

Order 48: $C_2\times C_{24}$ x 34, $C_2^2\times C_{12}$ x 21, $C_6\times D_4$ x 12, $C_2^2:C_{12}$ x 8, $C_4:C_{12}$ x 4, $C_4\times C_{12}$ x 4, $C_2^3\times C_6$ x 2

Order 32: $C_2^2\times C_8$ x 20, $C_2^2:C_8$ x 8, $C_4\times D_4$ x 8, $C_4\times C_8$ x 4, $C_4:C_8$ x 4, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 40, $C_2^2\times C_6$ x 21, $C_3\times D_4$ x 16, $C_{24}$ x 10

Order 16: $C_2\times C_8$ x 34, $C_2^2\times C_4$ x 21, $C_2\times D_4$ x 12, $C_2^2:C_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 39, $C_{12}$ x 14

Order 8: $C_2\times C_4$ x 40, $C_2^3$ x 21, $D_4$ x 16, $C_8$ x 10

Order 6: $C_6$ x 15

Order 4: $C_2^2$ x 39, $C_4$ x 14

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2\times D_4\times C_{24}$

Order 192: $D_4\times C_{24}$, $C_4^2.C_{12}$, $C_2^3:C_{24}$, $C_2\times C_4\times C_{24}$, $C_2^3\times C_{24}$, $C_2\times D_4\times C_{12}$

Order 128: $C_4^2.C_2^3$

Order 96: $C_2^2\times C_{24}$ x 5, $D_4\times C_{12}$ x 2, $C_4:C_{24}$, $C_2^2:C_{24}$, $C_4\times C_{24}$, $C_{12}:C_2^3$, $C_2^3\times C_{12}$, $C_2\times C_4:C_{12}$, $C_2^3:C_{12}$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^3:C_8$, $C_2\times C_4\times C_8$, $C_2^3\times C_8$, $C_4^2:C_2^2$, $C_8\times D_4$, $C_4^2.C_4$

Order 48: $C_2^2\times C_{12}$ x 8, $C_2\times C_{24}$ x 7, $C_6\times D_4$ x 2, $C_4\times C_{12}$ x 2, $C_2^3\times C_6$, $C_4:C_{12}$, $C_2^2:C_{12}$

Order 32: $C_2^2\times C_8$ x 5, $C_4\times D_4$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_4\times C_8$, $C_2^2.D_4$, $C_2^3:C_4$, $C_2\times C_4^2$, $C_4:C_8$

Order 24: $C_2\times C_{12}$ x 13, $C_2^2\times C_6$ x 5, $C_{24}$ x 3, $C_3\times D_4$

Order 16: $C_2^2\times C_4$ x 8, $C_2\times C_8$ x 7, $C_4^2$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$, $C_2^2:C_4$, $C_2^4$

Order 12: $C_2\times C_6$ x 8, $C_{12}$ x 6

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 5, $C_8$ x 3, $D_4$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 8, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2\times D_4\times C_{24}$ ($C_1$)

Order 192: $D_4\times C_{24}$ ($C_2$) x 8, $C_2^3:C_{24}$ ($C_2$) x 2, $C_2^3\times C_{24}$ ($C_2$) x 2, $C_4^2.C_{12}$ ($C_2$), $C_2\times C_4\times C_{24}$ ($C_2$), $C_2\times D_4\times C_{12}$ ($C_2$)

Order 128: $C_4^2.C_2^3$ ($C_3$)

Order 96: $C_2^2\times C_{24}$ ($C_2^2$) x 12, $C_2^2:C_{24}$ ($C_2^2$) x 8, $C_4:C_{24}$ ($C_2^2$) x 4, $C_4\times C_{24}$ ($C_2^2$) x 4, $D_4\times C_{12}$ ($C_2^2$) x 4, $D_4\times C_{12}$ ($C_4$) x 4, $C_2^3\times C_{12}$ ($C_2^2$) x 2, $C_2^3:C_{12}$ ($C_4$) x 2, $C_{12}:C_2^3$ ($C_4$), $C_2\times C_4:C_{12}$ ($C_4$), $C_2\times C_4\times C_{12}$ ($C_2^2$)

Order 64: $C_8\times D_4$ ($C_6$) x 8, $C_2^3:C_8$ ($C_6$) x 2, $C_2^3\times C_8$ ($C_6$) x 2, $C_2\times C_4\times C_8$ ($C_6$), $C_4^2:C_2^2$ ($C_6$), $C_4^2.C_4$ ($C_6$)

Order 48: $C_6\times D_4$ ($C_8$) x 8, $C_2^2\times C_{12}$ ($C_2\times C_4$) x 8, $C_2\times C_{24}$ ($C_2^3$) x 8, $C_2^2:C_{12}$ ($C_2\times C_4$) x 8, $C_2^2\times C_{12}$ ($C_2^3$) x 5, $C_6\times D_4$ ($C_2\times C_4$) x 4, $C_2\times C_{24}$ ($D_4$) x 4, $C_4:C_{12}$ ($C_2\times C_4$) x 4, $C_2^3\times C_6$ ($C_2\times C_4$) x 2, $C_4\times C_{12}$ ($C_2^3$) x 2, $C_4\times C_{12}$ ($C_2\times C_4$) x 2

Order 32: $C_2^2\times C_8$ ($C_2\times C_6$) x 12, $C_2^2:C_8$ ($C_2\times C_6$) x 8, $C_4\times C_8$ ($C_2\times C_6$) x 4, $C_4\times D_4$ ($C_2\times C_6$) x 4, $C_4\times D_4$ ($C_{12}$) x 4, $C_4:C_8$ ($C_2\times C_6$) x 4, $C_2^3\times C_4$ ($C_2\times C_6$) x 2, $C_2^3:C_4$ ($C_{12}$) x 2, $C_2^2\times D_4$ ($C_{12}$), $C_2^2.D_4$ ($C_{12}$), $C_2\times C_4^2$ ($C_2\times C_6$)

Order 24: $C_3\times D_4$ ($C_2\times C_8$) x 16, $C_2\times C_{12}$ ($C_2^2\times C_4$) x 9, $C_2^2\times C_6$ ($C_2\times C_8$) x 8, $C_2^2\times C_6$ ($C_2^2\times C_4$) x 5, $C_2\times C_{12}$ ($C_2\times C_8$) x 4, $C_{24}$ ($C_2\times D_4$) x 4, $C_2\times C_{12}$ ($D_4:C_2$) x 2, $C_2\times C_{12}$ ($C_2\times D_4$) x 2, $C_2\times C_{12}$ ($C_2^4$)

Order 16: $C_2\times C_8$ ($C_2^2\times C_6$) x 8, $C_2^2:C_4$ ($C_2\times C_{12}$) x 8, $C_2\times D_4$ ($C_{24}$) x 8, $C_2^2\times C_4$ ($C_2\times C_{12}$) x 8, $C_2^2\times C_4$ ($C_2^2\times C_6$) x 5, $C_2\times C_8$ ($C_3\times D_4$) x 4, $C_4:C_4$ ($C_2\times C_{12}$) x 4, $C_2\times D_4$ ($C_2\times C_{12}$) x 4, $C_4^2$ ($C_2\times C_{12}$) x 2, $C_4^2$ ($C_2^2\times C_6$) x 2, $C_2^4$ ($C_2\times C_{12}$) x 2

Order 12: $C_2\times C_6$ ($C_2^2\times C_8$) x 10, $C_{12}$ ($C_2^2\times C_8$) x 4, $C_2\times C_6$ ($\OD_{16}:C_2$) x 2, $C_2\times C_6$ ($C_4\times D_4$) x 2, $C_{12}$ ($C_4\times D_4$) x 2, $C_2\times C_6$ ($C_2^3\times C_4$), $C_{12}$ ($D_4:C_2^2$), $C_{12}$ ($C_2^2\times D_4$)

Order 8: $D_4$ ($C_2\times C_{24}$) x 16, $C_2\times C_4$ ($C_2^2\times C_{12}$) x 9, $C_2^3$ ($C_2\times C_{24}$) x 8, $C_2^3$ ($C_2^2\times C_{12}$) x 5, $C_2\times C_4$ ($C_2\times C_{24}$) x 4, $C_8$ ($C_6\times D_4$) x 4, $C_2\times C_4$ ($D_4:C_6$) x 2, $C_2\times C_4$ ($C_6\times D_4$) x 2, $C_2\times C_4$ ($C_2^3\times C_6$)

Order 6: $C_6$ ($C_8\times D_4$) x 4, $C_6$ ($C_8.C_2^3$), $C_6$ ($C_2^3\times C_8$), $C_6$ ($C_4^2:C_2^2$)

Order 4: $C_2^2$ ($C_2^2\times C_{24}$) x 10, $C_4$ ($C_2^2\times C_{24}$) x 4, $C_2^2$ ($\OD_{16}:C_6$) x 2, $C_2^2$ ($D_4\times C_{12}$) x 2, $C_4$ ($D_4\times C_{12}$) x 2, $C_2^2$ ($C_2^3\times C_{12}$), $C_4$ ($C_6.C_2^4$), $C_4$ ($C_{12}:C_2^3$)

Order 3: $C_3$ ($C_4^2.C_2^3$)

Order 2: $C_2$ ($D_4\times C_{24}$) x 4, $C_2$ ($C_{12}.C_2^4$), $C_2$ ($C_2^3\times C_{24}$), $C_2$ ($C_2\times D_4\times C_{12}$)

Order 1: $C_1$ ($C_2\times D_4\times C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2\times D_4\times C_{24}$ ($C_1$)

Order 192: $D_4\times C_{24}$ ($C_2$), $C_4^2.C_{12}$ ($C_2$), $C_2^3:C_{24}$ ($C_2$), $C_2\times C_4\times C_{24}$ ($C_2$), $C_2^3\times C_{24}$ ($C_2$), $C_2\times D_4\times C_{12}$ ($C_2$)

Order 128: $C_4^2.C_2^3$ ($C_3$)

Order 96: $C_2^2\times C_{24}$ ($C_2^2$) x 4, $C_4:C_{24}$ ($C_2^2$), $C_2^2:C_{24}$ ($C_2^2$), $C_4\times C_{24}$ ($C_2^2$), $C_{12}:C_2^3$ ($C_4$), $C_2^3\times C_{12}$ ($C_2^2$), $D_4\times C_{12}$ ($C_2^2$), $D_4\times C_{12}$ ($C_4$), $C_2\times C_4:C_{12}$ ($C_4$), $C_2^3:C_{12}$ ($C_4$), $C_2\times C_4\times C_{12}$ ($C_2^2$)

Order 64: $C_2^3:C_8$ ($C_6$), $C_2\times C_4\times C_8$ ($C_6$), $C_2^3\times C_8$ ($C_6$), $C_4^2:C_2^2$ ($C_6$), $C_8\times D_4$ ($C_6$), $C_4^2.C_4$ ($C_6$)

Order 48: $C_2^2\times C_{12}$ ($C_2\times C_4$) x 4, $C_2\times C_{24}$ ($C_2^3$) x 3, $C_2^2\times C_{12}$ ($C_2^3$) x 2, $C_2^3\times C_6$ ($C_2\times C_4$), $C_6\times D_4$ ($C_2\times C_4$), $C_6\times D_4$ ($C_8$), $C_2\times C_{24}$ ($D_4$), $C_4:C_{12}$ ($C_2\times C_4$), $C_2^2:C_{12}$ ($C_2\times C_4$), $C_4\times C_{12}$ ($C_2^3$), $C_4\times C_{12}$ ($C_2\times C_4$)

Order 32: $C_2^2\times C_8$ ($C_2\times C_6$) x 4, $C_2^2:C_8$ ($C_2\times C_6$), $C_2^2\times D_4$ ($C_{12}$), $C_2^3\times C_4$ ($C_2\times C_6$), $C_4\times C_8$ ($C_2\times C_6$), $C_4\times D_4$ ($C_2\times C_6$), $C_4\times D_4$ ($C_{12}$), $C_2^2.D_4$ ($C_{12}$), $C_2^3:C_4$ ($C_{12}$), $C_2\times C_4^2$ ($C_2\times C_6$), $C_4:C_8$ ($C_2\times C_6$)

Order 24: $C_2\times C_{12}$ ($C_2^2\times C_4$) x 4, $C_2^2\times C_6$ ($C_2^2\times C_4$) x 2, $C_2\times C_{12}$ ($C_2\times C_8$), $C_2\times C_{12}$ ($C_2^4$), $C_2\times C_{12}$ ($D_4:C_2$), $C_2\times C_{12}$ ($C_2\times D_4$), $C_{24}$ ($C_2\times D_4$), $C_2^2\times C_6$ ($C_2\times C_8$), $C_3\times D_4$ ($C_2\times C_8$)

Order 16: $C_2^2\times C_4$ ($C_2\times C_{12}$) x 4, $C_2\times C_8$ ($C_2^2\times C_6$) x 3, $C_2^2\times C_4$ ($C_2^2\times C_6$) x 2, $C_2\times C_8$ ($C_3\times D_4$), $C_4:C_4$ ($C_2\times C_{12}$), $C_2^2:C_4$ ($C_2\times C_{12}$), $C_4^2$ ($C_2\times C_{12}$), $C_4^2$ ($C_2^2\times C_6$), $C_2^4$ ($C_2\times C_{12}$), $C_2\times D_4$ ($C_2\times C_{12}$), $C_2\times D_4$ ($C_{24}$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_8$) x 2, $C_2\times C_6$ ($C_2^3\times C_4$), $C_2\times C_6$ ($\OD_{16}:C_2$), $C_2\times C_6$ ($C_4\times D_4$), $C_{12}$ ($D_4:C_2^2$), $C_{12}$ ($C_2^2\times D_4$), $C_{12}$ ($C_2^2\times C_8$), $C_{12}$ ($C_4\times D_4$)

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

Order 6: $C_6$ ($C_8.C_2^3$), $C_6$ ($C_2^3\times C_8$), $C_6$ ($C_4^2:C_2^2$), $C_6$ ($C_8\times D_4$)

Order 4: $C_2^2$ ($C_2^2\times C_{24}$) x 2, $C_2^2$ ($C_2^3\times C_{12}$), $C_2^2$ ($\OD_{16}:C_6$), $C_2^2$ ($D_4\times C_{12}$), $C_4$ ($C_6.C_2^4$), $C_4$ ($C_{12}:C_2^3$), $C_4$ ($C_2^2\times C_{24}$), $C_4$ ($D_4\times C_{12}$)

Order 3: $C_3$ ($C_4^2.C_2^3$)

Order 2: $C_2$ ($D_4\times C_{24}$), $C_2$ ($C_{12}.C_2^4$), $C_2$ ($C_2^3\times C_{24}$), $C_2$ ($C_2\times D_4\times C_{12}$)

Order 1: $C_1$ ($C_2\times D_4\times C_{24}$)

Series

Derived series

$C_2\times D_4\times C_{24}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2\times D_4\times C_{24}$

$\rhd$

$C_2^3\times C_{24}$

$\rhd$

$C_2^3\times C_{12}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times D_4\times C_{24}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{24}$

$\lhd$

$C_2\times D_4\times C_{24}$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $240 \times 240$ 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 $80 \times 80$ rational character table.