Abstract group 192.891: $C_3\times C_2^3.D_4$

• Label: 192.891
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $19$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_3\times C_2^3.D_4$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^5:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$
Nilpotency class:	$3$
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	12
Elements	1	19	2	44	38	88	192
Conjugacy classes	1	7	2	8	14	16	48
Divisions	1	7	1	7	7	7	30
Autjugacy classes	1	3	1	3	3	3	14

Dimension	1	2	4	8	16
Irr. complex chars.	24	18	6	0	0	48
Irr. rational chars.	8	14	6	1	1	30

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$2912$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{12}=[a,b]=[a,c]=1, d^{a}=bc^{2}d, c^{b}=c^{3}, d^{b}=c^{2}d, d^{c}=d^{7} \rangle$
Permutation group:	Degree $19$ $\langle(1,13,2,14)(3,16,4,15)(5,12,6,11)(7,9,8,10), (1,9)(2,10)(3,11)(4,12)(5,13) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_2^3.D_4)$
Semidirect product:	$(C_2^2.D_4)$ $\,\rtimes\,$ $C_6$	$(C_2^3:C_4)$ $\,\rtimes\,$ $C_6$	$(C_2^3:C_{12})$ $\,\rtimes\,$ $C_2$	$(C_2^2\times C_{12})$ $\,\rtimes\,$ $C_2^2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{12})$ . $D_4$	$(C_6\times D_4)$ . $C_2^2$	$(D_4:C_2^2)$ . $C_6$	$(C_2^2\times C_6)$ . $D_4$	all 14

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

Homology

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

Subgroups

There are 322 subgroups in 160 conjugacy classes, 54 normal (12 characteristic).

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

Special subgroups

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

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 192: $C_3\times C_2^3.D_4$

Order 96: $C_2^3:C_{12}$ x 3, $(C_2^2\times C_{12}):C_2$ x 3, $C_6.C_2^4$

Order 64: $C_2^3.D_4$

Order 48: $C_6\times D_4$ x 6, $C_2^2:C_{12}$ x 6, $D_4:C_6$ x 3, $C_4:C_{12}$ x 3, $C_2^2\times C_{12}$

Order 32: $C_2^3:C_4$ x 3, $C_2^2.D_4$ x 3, $D_4:C_2^2$

Order 24: $C_2\times C_{12}$ x 12, $C_3\times D_4$ x 9, $C_2^2\times C_6$ x 5, $C_3\times Q_8$

Order 16: $C_2^2:C_4$ x 6, $C_2\times D_4$ x 6, $C_4:C_4$ x 3, $D_4:C_2$ x 3, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 11, $C_{12}$ x 7

Order 8: $C_2\times C_4$ x 12, $D_4$ x 9, $C_2^3$ x 5, $Q_8$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 11, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_3\times C_2^3.D_4$

Order 96: $C_2^3:C_{12}$, $C_6.C_2^4$, $(C_2^2\times C_{12}):C_2$

Order 64: $C_2^3.D_4$

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

Order 32: $C_2^3:C_4$, $D_4:C_2^2$, $C_2^2.D_4$

Order 24: $C_2\times C_{12}$ x 4, $C_2^2\times C_6$ x 3, $C_3\times D_4$ x 2, $C_3\times Q_8$

Order 16: $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$, $D_4:C_2$, $C_2^2\times C_4$

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

Order 8: $C_2\times C_4$ x 4, $C_2^3$ x 3, $D_4$ x 2, $Q_8$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_3\times C_2^3.D_4$ ($C_1$)

Order 96: $C_2^3:C_{12}$ ($C_2$) x 3, $(C_2^2\times C_{12}):C_2$ ($C_2$) x 3, $C_6.C_2^4$ ($C_2$)

Order 64: $C_2^3.D_4$ ($C_3$)

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

Order 32: $C_2^3:C_4$ ($C_6$) x 3, $C_2^2.D_4$ ($C_6$) x 3, $D_4:C_2^2$ ($C_6$)

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

Order 16: $C_2^2:C_4$ ($C_2\times C_6$) x 3, $C_2\times D_4$ ($C_2\times C_6$) x 3, $C_2^2\times C_4$ ($C_2\times C_6$)

Order 12: $C_2\times C_6$ ($C_2\times D_4$) x 3

Order 8: $C_2^3$ ($C_3\times D_4$) x 3, $C_2\times C_4$ ($C_3\times D_4$) x 3, $C_2^3$ ($C_2^2\times C_6$)

Order 6: $C_6$ ($C_2^2\wr C_2$)

Order 4: $C_2^2$ ($C_6\times D_4$) x 3

Order 3: $C_3$ ($C_2^3.D_4$)

Order 2: $C_2$ ($C_2^4:C_6$)

Order 1: $C_1$ ($C_3\times C_2^3.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_3\times C_2^3.D_4$ ($C_1$)

Order 96: $C_2^3:C_{12}$ ($C_2$), $C_6.C_2^4$ ($C_2$), $(C_2^2\times C_{12}):C_2$ ($C_2$)

Order 64: $C_2^3.D_4$ ($C_3$)

Order 48: $C_6\times D_4$ ($C_2^2$), $C_2^2\times C_{12}$ ($C_2^2$), $C_2^2:C_{12}$ ($C_2^2$)

Order 32: $C_2^3:C_4$ ($C_6$), $D_4:C_2^2$ ($C_6$), $C_2^2.D_4$ ($C_6$)

Order 24: $C_2\times C_{12}$ ($D_4$), $C_2^2\times C_6$ ($C_2^3$), $C_2^2\times C_6$ ($D_4$)

Order 16: $C_2^2:C_4$ ($C_2\times C_6$), $C_2\times D_4$ ($C_2\times C_6$), $C_2^2\times C_4$ ($C_2\times C_6$)

Order 12: $C_2\times C_6$ ($C_2\times D_4$)

Order 8: $C_2^3$ ($C_2^2\times C_6$), $C_2^3$ ($C_3\times D_4$), $C_2\times C_4$ ($C_3\times D_4$)

Order 6: $C_6$ ($C_2^2\wr C_2$)

Order 4: $C_2^2$ ($C_6\times D_4$)

Order 3: $C_3$ ($C_2^3.D_4$)

Order 2: $C_2$ ($C_2^4:C_6$)

Order 1: $C_1$ ($C_3\times C_2^3.D_4$)

Series

Derived series

$C_3\times C_2^3.D_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_3\times C_2^3.D_4$

$\rhd$

$C_6.C_2^4$

$\rhd$

$C_6\times D_4$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_3\times C_2^3.D_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_3\times C_2^3.D_4$

Supergroups

This group is a maximal subgroup of 38 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 $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.