Abstract group 192.142: $(C_2\times C_4):C_{24}$

• Label: 192.142
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $19$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$(C_2\times C_4):C_{24}$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^7:D_4$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 6, $C_3$
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	7	2	24	14	32	48	64	192
Conjugacy classes	1	7	2	16	14	16	32	32	120
Divisions	1	7	1	8	7	4	8	4	40
Autjugacy classes	1	5	1	3	5	1	3	1	20

Dimension	1	2	4	8
Irr. complex chars.	96	24	0	0	120
Irr. rational chars.	4	14	16	6	40

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{8}=c^{6}=[a,c]=[b,c]=1, b^{a}=bc^{3} \rangle$
Permutation group:	Degree $19$ $\langle(1,2,4,7)(3,6,5,8)(9,10,12,14,11,13,15,16), (1,2,5,8)(3,6,4,7)(9,11)(10,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 5 \\ 10 & 13 \end{array}\right), \left(\begin{array}{rr} 17 & 3 \\ 16 & 13 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 4 \\ 8 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_2^2.C_4^2)$
Semidirect product:	$(C_2\times C_{24})$ $\,\rtimes\,$ $C_4$ (4)	$(C_2\times C_{12})$ $\,\rtimes\,$ $C_8$ (4)	$(C_2\times C_8)$ $\,\rtimes\,$ $C_{12}$ (4)	$(C_2\times C_4)$ $\,\rtimes\,$ $C_{24}$ (4)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_4\times C_8)$	$C_6$ . $(C_8:C_4)$	$C_6$ . $(C_4:C_8)$ (2)	$(C_2\times C_{12})$ . $Q_8$	all 40

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

Homology

Abelianization:	$C_{4} \times C_{24} \simeq C_{4} \times C_{8} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 154 subgroups in 118 conjugacy classes, 82 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{12}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_4\times C_{24}$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2\times C_4):C_{24}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2\times C_4):C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.C_4^2$
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

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 192: $(C_2\times C_4):C_{24}$

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

Order 64: $C_2^2.C_4^2$

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

Order 32: $C_2^2\times C_8$ x 2, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 14, $C_{24}$ x 4, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 8, $C_2^2\times C_4$ x 3, $C_4^2$ x 2

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 14, $C_8$ x 4, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 8, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $(C_2\times C_4):C_{24}$

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

Order 64: $C_2^2.C_4^2$

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

Order 32: $C_2^2\times C_8$, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 7, $C_{24}$, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 3, $C_2^2\times C_4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 5, $C_{12}$ x 3

Order 8: $C_2\times C_4$ x 7, $C_2^3$, $C_8$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $(C_2\times C_4):C_{24}$ ($C_1$)

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

Order 64: $C_2^2.C_4^2$ ($C_3$)

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

Order 32: $C_2^2\times C_8$ ($C_6$) x 2, $C_2\times C_4^2$ ($C_6$)

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

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

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

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

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

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

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

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

Order 1: $C_1$ ($(C_2\times C_4):C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $(C_2\times C_4):C_{24}$ ($C_1$)

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

Order 64: $C_2^2.C_4^2$ ($C_3$)

Order 48: $C_2^2\times C_{12}$ ($C_2^2$), $C_2^2\times C_{12}$ ($C_4$), $C_2\times C_{24}$ ($C_4$)

Order 32: $C_2^2\times C_8$ ($C_6$), $C_2\times C_4^2$ ($C_6$)

Order 24: $C_2\times C_{12}$ ($D_4$) x 2, $C_2\times C_{12}$ ($Q_8$), $C_2\times C_{12}$ ($C_2\times C_4$), $C_2\times C_{12}$ ($C_8$), $C_2^2\times C_6$ ($C_2\times C_4$)

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

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

Order 8: $C_2\times C_4$ ($C_3\times D_4$) x 2, $C_2^3$ ($C_2\times C_{12}$), $C_2\times C_4$ ($C_2\times C_{12}$), $C_2\times C_4$ ($C_{24}$), $C_2\times C_4$ ($C_3\times Q_8$)

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

Order 4: $C_2^2$ ($C_3\times \OD_{16}$), $C_2^2$ ($C_2\times C_{24}$), $C_2^2$ ($C_4:C_{12}$), $C_2^2$ ($C_2^2:C_{12}$), $C_2^2$ ($C_4\times C_{12}$), $C_4$ ($C_4:C_{12}$), $C_4$ ($C_2^2:C_{12}$)

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

Order 2: $C_2$ ($C_4:C_{24}$), $C_2$ ($C_2^2:C_{24}$), $C_2$ ($C_8:C_{12}$), $C_2$ ($C_4\times C_{24}$), $C_2$ ($C_6.C_4^2$)

Order 1: $C_1$ ($(C_2\times C_4):C_{24}$)

Series

Derived series

$(C_2\times C_4):C_{24}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$(C_2\times C_4):C_{24}$

$\rhd$

$C_2\times C_4\times C_{12}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_4):C_{24}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{12}$

$\lhd$

$(C_2\times C_4):C_{24}$

Supergroups

This group is a maximal subgroup of 221 larger groups in the database.

This group is a maximal quotient of 39 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.