Abstract group 384.19851: $(C_2^2\times D_4):D_6$

• Label: 384.19851
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $5$

Group information

Description:	$(C_2^2\times D_4):D_6$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$
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	12
Elements	1	103	2	152	62	64	384
Conjugacy classes	1	20	1	23	19	20	84
Divisions	1	20	1	22	14	12	70
Autjugacy classes	1	7	1	7	6	4	26

Dimension	1	2	4	8	16
Irr. complex chars.	32	40	12	0	0	84
Irr. rational chars.	32	24	10	3	1	70

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$5$
Inequivalent generating 5-tuples:	$33331200$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{4}=e^{12}=[a,b]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(3,5)(4,6)(7,8)(11,12)(14,15), (1,3)(2,7)(4,8)(5,6)(9,10)(11,12), (1,4) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$Q_8$ $\,\rtimes\,$ $(C_6:D_4)$	$(D_4:C_6)$ $\,\rtimes\,$ $D_4$	$D_4$ $\,\rtimes\,$ $(C_6:D_4)$	$(D_4:C_2^3)$ $\,\rtimes\,$ $S_3$	all 51
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:Q_8)$ . $C_2^3$	$(C_6\times Q_8)$ . $C_2^3$ (2)	$(D_4:C_2^2)$ . $D_6$	$(C_6:D_4)$ . $C_2^3$ (4)	all 40

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 4766 subgroups in 1658 conjugacy classes, 553 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^3\times D_6$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.C_2^4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2^2\times D_4):D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 384: $(C_2^2\times D_4):D_6$

Order 192: $(C_2\times D_4):D_6$ x 6, $C_2^4:D_6$ x 6, $(C_2\times C_{12}):D_4$ x 3, $(C_2\times C_{12}):D_4$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $(C_3\times Q_8):D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $C_{12}.(C_2^2\times C_4)$

Order 128: $D_4^2:C_2$

Order 96: $C_{12}:D_4$ x 24, $C_6.C_2^4$ x 16, $C_2^3.D_6$ x 12, $D_6:D_4$ x 12, $C_{12}:D_4$ x 12, $D_4:D_6$ x 8, $C_2^3:D_6$ x 6, $D_4\times D_6$ x 6, $C_2^4:S_3$ x 6, $C_{12}.D_4$ x 6, $D_6:Q_8$ x 6, $C_{12}:D_4$ x 6, $C_{12}:D_4$ x 6, $C_2^3.D_6$ x 6, $C_2^3.D_6$ x 6, $C_2^2.D_{12}$ x 6, $C_{12}.D_4$ x 4, $C_{12}:C_2^3$ x 3, $C_2^3\times C_{12}$ x 3, $D_{12}:C_2^2$ x 3, $C_2^2\times D_{12}$ x 3, $C_2^3.D_6$ x 3, $C_2^2.D_{12}$ x 3, $Q_8:D_6$ x 2, $C_3:C_4\times Q_8$ x 2, $C_6.C_2^4$

Order 64: $D_4:D_4$ x 6, $D_4^2$ x 6, $C_2^2.C_2^4$ x 3, $C_4^2:C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3:D_4$ x 3, $Q_8:D_4$ x 2, $Q_8:D_4$ x 2, $D_4:C_2^3$, $D_4:C_2^3$, $C_4^2.C_2^2$

Order 48: $C_6:D_4$ x 42, $D_4:C_6$ x 36, $C_2^2\times C_{12}$ x 28, $C_6\times D_4$ x 24, $S_3\times D_4$ x 24, $D_6:C_4$ x 24, $C_2\times D_{12}$ x 21, $C_6.D_4$ x 14, $D_{12}:C_2$ x 12, $C_{12}:C_4$ x 10, $C_6\times Q_8$ x 8, $D_{12}:C_2$ x 8, $C_6.D_4$ x 8, $C_2^2\times D_6$ x 6, $C_6.C_2^3$ x 6, $C_4\times D_6$ x 6, $C_{12}:C_4$ x 6, $C_2^3\times C_6$ x 3, $C_6:Q_8$

Order 32: $C_4:D_4$ x 42, $D_4:C_2^2$ x 21, $C_2^2\times D_4$ x 18, $C_2^2\wr C_2$ x 18, $C_4\times D_4$ x 18, $C_2^2:Q_8$ x 10, $D_4:C_2^2$ x 8, $C_4:D_4$ x 6, $C_4^2:C_2$ x 6, $C_2^2.D_4$ x 6, $C_2^3:C_4$ x 6, $C_2^3\times C_4$ x 3, $C_4^2:C_2$ x 3, $C_2^2.D_4$ x 3, $C_4\times Q_8$ x 2, $C_2^2\times Q_8$

Order 24: $C_2\times C_{12}$ x 46, $C_3:D_4$ x 42, $C_2\times D_6$ x 30, $C_3\times D_4$ x 30, $C_2^2\times C_6$ x 22, $D_{12}$ x 18, $C_6:C_4$ x 16, $C_4\times S_3$ x 12, $C_3\times Q_8$ x 10, $C_3:Q_8$ x 2

Order 16: $C_2\times D_4$ x 111, $D_4:C_2$ x 56, $C_2^2\times C_4$ x 40, $C_2^2:C_4$ x 38, $C_4:C_4$ x 18, $C_2^4$ x 9, $C_2\times Q_8$ x 9, $C_4^2$ x 6

Order 12: $C_2\times C_6$ x 38, $D_6$ x 30, $C_{12}$ x 12, $C_3:C_4$ x 10

Order 8: $D_4$ x 90, $C_2\times C_4$ x 74, $C_2^3$ x 52, $Q_8$ x 12

Order 6: $C_6$ x 14, $S_3$ x 6

Order 4: $C_2^2$ x 68, $C_4$ x 22

Order 3: $C_3$

Order 2: $C_2$ x 20

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_2^2\times D_4):D_6$

Order 192: $C_{12}.C_2^4$, $D_{12}:C_2^3$, $(C_3\times Q_8):D_4$, $(C_2\times C_{12}):D_4$, $(C_2\times C_{12}):D_4$, $(C_2\times D_4):D_6$, $C_{12}.(C_2^2\times C_4)$, $(C_3\times Q_8):D_4$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$

Order 128: $D_4^2:C_2$

Order 96: $C_6.C_2^4$ x 5, $C_{12}:D_4$ x 5, $C_{12}:D_4$ x 2, $C_{12}.D_4$ x 2, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_2^3\times C_{12}$, $C_2^3:D_6$, $D_4:D_6$, $Q_8:D_6$, $D_4\times D_6$, $D_{12}:C_2^2$, $C_2^2\times D_{12}$, $C_2^4:S_3$, $C_{12}.D_4$, $D_6:Q_8$, $C_3:C_4\times Q_8$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $D_6:D_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^2.D_{12}$, $C_2^3.D_6$, $C_2^2.D_{12}$

Order 64: $D_4:C_2^3$, $D_4:C_2^3$, $Q_8:D_4$, $Q_8:D_4$, $D_4:D_4$, $D_4^2$, $C_2^2.C_2^4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2.C_2^2$

Order 48: $C_2^2\times C_{12}$ x 7, $C_6:D_4$ x 5, $D_4:C_6$ x 4, $C_6\times D_4$ x 4, $C_6\times Q_8$ x 3, $C_2\times D_{12}$ x 3, $C_6.D_4$ x 3, $D_6:C_4$ x 3, $C_{12}:C_4$ x 3, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_2^2\times D_6$, $C_6.C_2^3$, $D_{12}:C_2$, $S_3\times D_4$, $D_{12}:C_2$, $C_4\times D_6$, $C_6:Q_8$, $C_{12}:C_4$

Order 32: $D_4:C_2^2$ x 7, $C_4:D_4$ x 7, $C_2^2\times D_4$ x 4, $C_2^2:Q_8$ x 3, $C_4\times D_4$ x 3, $C_2^2\wr C_2$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2^3\times C_4$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2.D_4$, $C_4\times Q_8$, $C_4^2:C_2$, $C_2^2.D_4$, $C_2^3:C_4$

Order 24: $C_2\times C_{12}$ x 10, $C_2^2\times C_6$ x 6, $C_3:D_4$ x 4, $C_6:C_4$ x 4, $C_2\times D_6$ x 4, $C_3\times D_4$ x 3, $D_{12}$ x 2, $C_3\times Q_8$ x 2, $C_4\times S_3$, $C_3:Q_8$

Order 16: $C_2\times D_4$ x 13, $C_2^2\times C_4$ x 9, $C_2^2:C_4$ x 6, $D_4:C_2$ x 6, $C_4:C_4$ x 5, $C_2\times Q_8$ x 4, $C_2^4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 11, $D_6$ x 4, $C_{12}$ x 4, $C_3:C_4$ x 3

Order 8: $C_2\times C_4$ x 15, $C_2^3$ x 10, $D_4$ x 9, $Q_8$ x 3

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

Order 4: $C_2^2$ x 15, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_2^2\times D_4):D_6$ ($C_1$)

Order 192: $(C_2\times D_4):D_6$ ($C_2$) x 6, $C_2^4:D_6$ ($C_2$) x 6, $(C_2\times C_{12}):D_4$ ($C_2$) x 3, $(C_2\times C_{12}):D_4$ ($C_2$) x 3, $C_2^4.D_6$ ($C_2$) x 3, $C_2^3:D_{12}$ ($C_2$) x 3, $(C_3\times Q_8):D_4$ ($C_2$) x 2, $(C_3\times Q_8):D_4$ ($C_2$) x 2, $C_{12}.C_2^4$ ($C_2$), $D_{12}:C_2^3$ ($C_2$), $C_{12}.(C_2^2\times C_4)$ ($C_2$)

Order 96: $C_{12}:D_4$ ($C_2^2$) x 24, $C_2^3.D_6$ ($C_2^2$) x 12, $D_6:D_4$ ($C_2^2$) x 12, $C_{12}:D_4$ ($C_2^2$) x 12, $C_6.C_2^4$ ($C_2^2$) x 8, $C_2^3:D_6$ ($C_2^2$) x 6, $D_4\times D_6$ ($C_2^2$) x 6, $C_2^4:S_3$ ($C_2^2$) x 6, $C_{12}.D_4$ ($C_2^2$) x 6, $D_6:Q_8$ ($C_2^2$) x 6, $C_{12}:D_4$ ($C_2^2$) x 6, $C_{12}:D_4$ ($C_2^2$) x 6, $C_2^3.D_6$ ($C_2^2$) x 6, $C_2^3.D_6$ ($C_2^2$) x 6, $C_2^2.D_{12}$ ($C_2^2$) x 6, $C_{12}.D_4$ ($C_2^2$) x 4, $C_{12}:C_2^3$ ($C_2^2$) x 3, $C_2^3\times C_{12}$ ($C_2^2$) x 3, $D_{12}:C_2^2$ ($C_2^2$) x 3, $C_2^2\times D_{12}$ ($C_2^2$) x 3, $C_2^3.D_6$ ($C_2^2$) x 3, $C_2^2.D_{12}$ ($C_2^2$) x 3, $Q_8:D_6$ ($C_2^2$) x 2, $C_3:C_4\times Q_8$ ($C_2^2$) x 2, $C_6.C_2^4$ ($C_2^2$)

Order 64: $D_4:C_2^3$ ($S_3$)

Order 48: $C_6:D_4$ ($C_2^3$) x 30, $D_6:C_4$ ($C_2^3$) x 24, $C_2^2\times C_{12}$ ($C_2^3$) x 16, $C_6.D_4$ ($C_2^3$) x 14, $C_6\times D_4$ ($C_2^3$) x 12, $C_{12}:C_4$ ($C_2^3$) x 10, $C_2\times D_{12}$ ($C_2^3$) x 9, $D_4:C_6$ ($D_4$) x 8, $C_6.D_4$ ($C_2^3$) x 8, $C_2^2\times D_6$ ($C_2^3$) x 6, $C_6.C_2^3$ ($C_2^3$) x 6, $C_4\times D_6$ ($C_2^3$) x 6, $C_{12}:C_4$ ($C_2^3$) x 6, $C_6\times Q_8$ ($C_2^3$) x 4, $C_2^3\times C_6$ ($C_2^3$) x 3, $C_6:Q_8$ ($C_2^3$)

Order 32: $D_4:C_2^2$ ($D_6$) x 8, $C_2^2\times D_4$ ($D_6$) x 3, $C_2^3\times C_4$ ($D_6$) x 3, $C_2^2\times Q_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($C_2\times D_4$) x 12, $C_3\times D_4$ ($C_2\times D_4$) x 12, $C_6:C_4$ ($C_2^4$) x 10, $C_2\times C_{12}$ ($C_2^4$) x 8, $C_2^2\times C_6$ ($C_2^4$) x 7, $C_2\times D_6$ ($C_2^4$) x 6, $C_3\times Q_8$ ($C_2\times D_4$) x 4

Order 16: $C_2^2\times C_4$ ($C_2\times D_6$) x 16, $C_2\times D_4$ ($C_2\times D_6$) x 12, $D_4:C_2$ ($C_3:D_4$) x 8, $C_2\times Q_8$ ($C_2\times D_6$) x 4, $C_2^4$ ($C_2\times D_6$) x 3

Order 12: $C_{12}$ ($C_2^2\times D_4$) x 8, $C_2\times C_6$ ($C_2^2\times D_4$) x 6, $C_2\times C_6$ ($D_4:C_2^2$) x 2, $C_2\times C_6$ ($C_2^5$)

Order 8: $D_4$ ($C_6:D_4$) x 12, $C_2\times C_4$ ($C_6:D_4$) x 12, $C_2\times C_4$ ($C_2^2\times D_6$) x 8, $C_2^3$ ($C_2^2\times D_6$) x 7, $Q_8$ ($C_6:D_4$) x 4

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

Order 4: $C_4$ ($C_2^3:D_6$) x 8, $C_2^2$ ($C_2^3:D_6$) x 6, $C_2^2$ ($D_4:D_6$) x 2, $C_2^2$ ($C_2^3\times D_6$)

Order 3: $C_3$ ($D_4^2:C_2$)

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

Order 1: $C_1$ ($(C_2^2\times D_4):D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_2^2\times D_4):D_6$ ($C_1$)

Order 192: $C_{12}.C_2^4$ ($C_2$), $D_{12}:C_2^3$ ($C_2$), $(C_3\times Q_8):D_4$ ($C_2$), $(C_2\times C_{12}):D_4$ ($C_2$), $(C_2\times C_{12}):D_4$ ($C_2$), $(C_2\times D_4):D_6$ ($C_2$), $C_{12}.(C_2^2\times C_4)$ ($C_2$), $(C_3\times Q_8):D_4$ ($C_2$), $C_2^4:D_6$ ($C_2$), $C_2^4.D_6$ ($C_2$), $C_2^3:D_{12}$ ($C_2$)

Order 96: $C_{12}:D_4$ ($C_2^2$) x 5, $C_6.C_2^4$ ($C_2^2$) x 4, $C_{12}:D_4$ ($C_2^2$) x 2, $C_{12}.D_4$ ($C_2^2$) x 2, $C_6.C_2^4$ ($C_2^2$), $C_{12}:C_2^3$ ($C_2^2$), $C_2^3\times C_{12}$ ($C_2^2$), $C_2^3:D_6$ ($C_2^2$), $Q_8:D_6$ ($C_2^2$), $D_4\times D_6$ ($C_2^2$), $D_{12}:C_2^2$ ($C_2^2$), $C_2^2\times D_{12}$ ($C_2^2$), $C_2^4:S_3$ ($C_2^2$), $C_{12}.D_4$ ($C_2^2$), $D_6:Q_8$ ($C_2^2$), $C_3:C_4\times Q_8$ ($C_2^2$), $C_{12}:D_4$ ($C_2^2$), $C_2^3.D_6$ ($C_2^2$), $C_{12}:D_4$ ($C_2^2$), $D_6:D_4$ ($C_2^2$), $C_2^3.D_6$ ($C_2^2$), $C_2^3.D_6$ ($C_2^2$), $C_2^2.D_{12}$ ($C_2^2$), $C_2^3.D_6$ ($C_2^2$), $C_2^2.D_{12}$ ($C_2^2$)

Order 64: $D_4:C_2^3$ ($S_3$)

Order 48: $C_2^2\times C_{12}$ ($C_2^3$) x 6, $C_6:D_4$ ($C_2^3$) x 4, $C_6\times D_4$ ($C_2^3$) x 3, $C_6.D_4$ ($C_2^3$) x 3, $D_6:C_4$ ($C_2^3$) x 3, $C_{12}:C_4$ ($C_2^3$) x 3, $C_6\times Q_8$ ($C_2^3$) x 2, $C_2\times D_{12}$ ($C_2^3$) x 2, $C_6.D_4$ ($C_2^3$) x 2, $C_2^3\times C_6$ ($C_2^3$), $C_2^2\times D_6$ ($C_2^3$), $D_4:C_6$ ($D_4$), $C_6.C_2^3$ ($C_2^3$), $C_4\times D_6$ ($C_2^3$), $C_6:Q_8$ ($C_2^3$), $C_{12}:C_4$ ($C_2^3$)

Order 32: $D_4:C_2^2$ ($D_6$) x 4, $C_2^2\times Q_8$ ($D_6$), $C_2^2\times D_4$ ($D_6$), $C_2^3\times C_4$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($C_2^4$) x 4, $C_6:C_4$ ($C_2^4$) x 3, $C_2^2\times C_6$ ($C_2^4$) x 3, $C_2\times C_{12}$ ($C_2\times D_4$), $C_2\times D_6$ ($C_2^4$), $C_3\times Q_8$ ($C_2\times D_4$), $C_3\times D_4$ ($C_2\times D_4$)

Order 16: $C_2^2\times C_4$ ($C_2\times D_6$) x 6, $C_2\times D_4$ ($C_2\times D_6$) x 3, $C_2\times Q_8$ ($C_2\times D_6$) x 2, $C_2^4$ ($C_2\times D_6$), $D_4:C_2$ ($C_3:D_4$)

Order 12: $C_{12}$ ($C_2^2\times D_4$) x 2, $C_2\times C_6$ ($C_2^5$), $C_2\times C_6$ ($D_4:C_2^2$), $C_2\times C_6$ ($C_2^2\times D_4$)

Order 8: $C_2\times C_4$ ($C_2^2\times D_6$) x 4, $C_2^3$ ($C_2^2\times D_6$) x 3, $Q_8$ ($C_6:D_4$), $D_4$ ($C_6:D_4$), $C_2\times C_4$ ($C_6:D_4$)

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

Order 4: $C_4$ ($C_2^3:D_6$) x 2, $C_2^2$ ($C_2^3\times D_6$), $C_2^2$ ($C_2^3:D_6$), $C_2^2$ ($D_4:D_6$)

Order 3: $C_3$ ($D_4^2:C_2$)

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

Order 1: $C_1$ ($(C_2^2\times D_4):D_6$)

Series

Derived series

$(C_2^2\times D_4):D_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$(C_2^2\times D_4):D_6$

$\rhd$

$C_2^4:D_6$

$\rhd$

$C_{12}:C_2^3$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times D_4):D_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$D_4:C_2^3$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $70 \times 70$ rational character table.