Abstract group 1152.155935: $C_{12}.\GL(2,\mathbb{Z}/4)$

• Label: 1152.155935
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_{12}.\GL(2,\mathbb{Z}/4)$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^2.D_6^2.C_2^4$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	135	26	88	342	288	272	1152
Conjugacy classes	1	10	3	7	17	4	18	60
Divisions	1	10	3	7	12	4	11	48
Autjugacy classes	1	8	3	7	11	2	11	43

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	8	18	8	13	10	0	3	0	0	60
Irr. rational chars.	8	10	8	9	6	2	3	1	1	48

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$40320$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=c^{12}=d^{2}=e^{6}=[a,d]=[d,e]=1, b^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,5)(3,7)(6,8)(10,11), (1,2)(3,8)(4,6)(5,7)(13,15), (2,6)(5,8), (1,3,4,7) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4:S_4)$ $\,\rtimes\,$ $D_6$	$(A_4:Q_8)$ $\,\rtimes\,$ $D_6$	$A_4$ $\,\rtimes\,$ $(D_4:D_6)$	$(D_{12}:S_4)$ $\,\rtimes\,$ $C_2$ (2)	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(D_6:S_4)$	$C_4$ . $(D_6\times S_4)$	$D_{12}$ . $(C_2\times S_4)$ (2)	$(C_{12}\times A_4)$ . $D_4$	all 33

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 6646 subgroups in 732 conjugacy classes, 59 normal (47 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6:\GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $C_{12}\times A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{12}$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_{12}.\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_6:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1152: $C_{12}.\GL(2,\mathbb{Z}/4)$

Order 576: $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $D_{12}:S_4$ x 2, $C_2\times A_4\times D_{12}$, $\GL(2,\mathbb{Z}/4):C_6$, $C_{12}.(C_2\times S_4)$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$, $(C_2\times D_{12}):D_4$

Order 288: $A_4\times D_{12}$ x 3, $C_{12}.S_4$ x 2, $C_2^3.C_6^2$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}.S_4$, $D_{12}:D_6$, $C_2\times A_4\times D_6$

Order 192: $D_{12}.D_4$ x 4, $D_{12}:D_4$ x 4, $D_4:S_4$ x 2, $A_4:\SD_{16}$ x 2, $A_4:\OD_{16}$ x 2, $(C_2\times D_{12}):C_4$ x 2, $C_{12}.C_2^4$ x 2, $(C_2\times C_6):\OD_{16}$, $C_2^3\times D_{12}$, $C_2^5:C_6$, $\GL(2,\mathbb{Z}/4):C_2$, $C_3\times C_4^2:C_2^2$

Order 144: $A_4\times D_6$ x 4, $C_{12}.D_6$ x 2, $D_{12}:S_3$ x 2, $C_{12}\times A_4$ x 2, $C_{12}.D_6$, $C_2^2:C_6^2$, $C_6\times S_4$, $D_{12}:C_6$, $C_6\times D_{12}$, $A_4:C_{12}$

Order 128: $C_2^4.D_4$

Order 96: $C_2^2\times D_{12}$ x 9, $D_{12}:C_4$ x 8, $D_4:D_6$ x 8, $A_4:C_8$ x 4, $C_{12}.D_4$ x 4, $Q_8:D_6$ x 4, $C_6:D_8$ x 4, $D_4\times A_4$ x 3, $C_2^3:C_{12}$ x 2, $C_{12}.D_4$ x 2, $C_{12}:D_4$ x 2, $D_4\times C_{12}$ x 2, $C_6:\OD_{16}$ x 2, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_6.C_2^4$, $C_2^3\times C_{12}$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $C_4\times S_4$, $A_4:Q_8$, $(C_2^2\times C_{12}):C_2$, $C_2^4:C_6$, $C_4^2:C_6$, $D_{12}:C_2^2$

Order 72: $C_3\times D_{12}$ x 4, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_3^2:C_8$ x 2, $C_6\times D_6$, $C_3\times S_4$, $C_6\times C_{12}$, $C_6\wr C_2$, $S_3\times C_{12}$, $C_3^2:Q_8$

Order 64: $C_2^2:\SD_{16}$ x 4, $D_4:D_4$ x 4, $C_2^3.D_4$ x 2, $C_8:C_2^3$ x 2, $C_2^2:\OD_{16}$, $D_4\times C_2^3$, $C_4^2:C_2^2$

Order 48: $C_2\times D_{12}$ x 25, $C_2^2\times D_6$ x 10, $C_3:D_8$ x 10, $Q_8:S_3$ x 8, $C_2^2\times C_{12}$ x 7, $C_2^2\times A_4$ x 6, $C_3:\OD_{16}$ x 6, $C_6\times D_4$ x 5, $C_2^2:C_{12}$ x 5, $C_6:C_8$ x 4, $D_4:C_6$ x 4, $C_4\times A_4$ x 4, $C_4:C_{12}$ x 4, $C_3:\SD_{16}$ x 2, $C_2^3\times C_6$, $C_2\times S_4$, $C_6\times Q_8$, $D_{12}:C_2$, $A_4:C_4$, $C_4\times C_{12}$

Order 36: $C_6\times S_3$ x 5, $C_3\times C_{12}$ x 2, $C_3:C_{12}$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 9, $D_4:C_4$ x 8, $D_8:C_2$ x 8, $C_2^2:C_8$ x 4, $C_2\times \SD_{16}$ x 4, $C_2\times D_8$ x 4, $C_2\times \OD_{16}$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $C_2^5$, $D_4:C_2^2$, $C_2^3\times C_4$, $C_2^2.D_4$, $C_2^2\wr C_2$, $C_4^2:C_2$

Order 24: $C_2\times D_6$ x 30, $D_{12}$ x 17, $C_2\times C_{12}$ x 16, $C_3\times D_4$ x 11, $C_3:C_8$ x 8, $C_2^2\times C_6$ x 7, $C_2\times A_4$ x 6, $C_3\times Q_8$ x 2, $C_3:D_4$, $C_4\times S_3$, $C_3:Q_8$, $S_4$

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 29, $C_2^4$ x 11, $\SD_{16}$ x 8, $D_8$ x 8, $C_2^2\times C_4$ x 7, $C_2^2:C_4$ x 5, $\OD_{16}$ x 4, $C_2\times C_8$ x 4, $C_4:C_4$ x 4, $D_4:C_2$ x 4, $C_4^2$, $C_2\times Q_8$

Order 12: $D_6$ x 25, $C_2\times C_6$ x 19, $C_{12}$ x 11, $A_4$ x 2, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 36, $D_4$ x 24, $C_2\times C_4$ x 14, $C_8$ x 4, $Q_8$ x 2

Order 6: $C_6$ x 12, $S_3$ x 5

Order 4: $C_2^2$ x 37, $C_4$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $C_{12}.\GL(2,\mathbb{Z}/4)$

Order 576: $C_2\times A_4\times D_{12}$, $\GL(2,\mathbb{Z}/4):C_6$, $C_{12}.(C_2\times S_4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$, $(C_2\times D_{12}):D_4$

Order 288: $A_4\times D_{12}$ x 2, $C_2^3.C_6^2$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}.S_4$, $D_{12}:D_6$, $C_{12}.S_4$, $C_2\times A_4\times D_6$

Order 192: $A_4:\OD_{16}$ x 2, $D_{12}.D_4$ x 2, $D_{12}:D_4$ x 2, $(C_2\times D_{12}):C_4$ x 2, $C_{12}.C_2^4$ x 2, $D_4:S_4$, $A_4:\SD_{16}$, $(C_2\times C_6):\OD_{16}$, $C_2^3\times D_{12}$, $C_2^5:C_6$, $\GL(2,\mathbb{Z}/4):C_2$, $C_3\times C_4^2:C_2^2$

Order 144: $A_4\times D_6$ x 2, $C_{12}\times A_4$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $C_2^2:C_6^2$, $C_6\times S_4$, $D_{12}:C_6$, $C_6\times D_{12}$, $A_4:C_{12}$

Order 128: $C_2^4.D_4$

Order 96: $C_2^2\times D_{12}$ x 7, $D_4:D_6$ x 5, $D_{12}:C_4$ x 4, $A_4:C_8$ x 2, $C_{12}.D_4$ x 2, $D_4\times A_4$ x 2, $C_2^3:C_{12}$ x 2, $C_{12}.D_4$ x 2, $C_{12}:D_4$ x 2, $D_4\times C_{12}$ x 2, $Q_8:D_6$ x 2, $C_6:D_8$ x 2, $C_6:\OD_{16}$ x 2, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_6.C_2^4$, $C_2^3\times C_{12}$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $C_4\times S_4$, $A_4:Q_8$, $(C_2^2\times C_{12}):C_2$, $C_2^4:C_6$, $C_4^2:C_6$, $D_{12}:C_2^2$

Order 72: $C_3\times D_{12}$ x 3, $C_6\times A_4$ x 2, $C_6\times D_6$, $S_3\times A_4$, $C_3\times S_4$, $C_6\times C_{12}$, $C_6\wr C_2$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:C_8$

Order 64: $C_2^3.D_4$ x 2, $C_8:C_2^3$ x 2, $C_2^2:\SD_{16}$ x 2, $D_4:D_4$ x 2, $C_2^2:\OD_{16}$, $D_4\times C_2^3$, $C_4^2:C_2^2$

Order 48: $C_2\times D_{12}$ x 15, $C_2^2\times C_{12}$ x 7, $C_2^2\times D_6$ x 6, $C_3:\OD_{16}$ x 6, $C_6\times D_4$ x 5, $C_2^2:C_{12}$ x 5, $C_2^2\times A_4$ x 4, $D_4:C_6$ x 4, $C_4\times A_4$ x 4, $C_4:C_{12}$ x 4, $C_3:D_8$ x 4, $Q_8:S_3$ x 3, $C_6:C_8$ x 2, $C_2^3\times C_6$, $C_2\times S_4$, $C_6\times Q_8$, $D_{12}:C_2$, $A_4:C_4$, $C_4\times C_{12}$, $C_3:\SD_{16}$

Order 36: $C_6\times S_3$ x 3, $C_3\times C_{12}$ x 2, $C_3:C_{12}$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 7, $D_8:C_2$ x 5, $D_4:C_4$ x 4, $C_2^2:C_8$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $C_2^5$, $D_4:C_2^2$, $C_2^3\times C_4$, $C_2^2.D_4$, $C_2^2\wr C_2$, $C_4^2:C_2$

Order 24: $C_2\times C_{12}$ x 16, $C_2\times D_6$ x 14, $D_{12}$ x 9, $C_3\times D_4$ x 9, $C_2^2\times C_6$ x 7, $C_2\times A_4$ x 5, $C_3:C_8$ x 4, $C_3\times Q_8$ x 2, $C_3:D_4$, $C_4\times S_3$, $C_3:Q_8$, $S_4$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$ x 19, $C_2^4$ x 7, $C_2^2\times C_4$ x 7, $C_2^2:C_4$ x 5, $\OD_{16}$ x 4, $C_4:C_4$ x 4, $D_4:C_2$ x 4, $\SD_{16}$ x 3, $D_8$ x 3, $C_2\times C_8$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $C_2\times C_6$ x 16, $D_6$ x 11, $C_{12}$ x 11, $A_4$ x 2, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 20, $D_4$ x 15, $C_2\times C_4$ x 14, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 11, $S_3$ x 3

Order 4: $C_2^2$ x 22, $C_4$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $C_{12}.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 576: $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $D_{12}:S_4$ ($C_2$) x 2, $C_2\times A_4\times D_{12}$ ($C_2$), $\GL(2,\mathbb{Z}/4):C_6$ ($C_2$), $C_{12}.(C_2\times S_4)$ ($C_2$)

Order 288: $A_4\times D_{12}$ ($C_2^2$) x 2, $C_{12}.S_4$ ($C_2^2$) x 2, $C_2^3.C_6^2$ ($C_2^2$), $C_{12}:S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$)

Order 192: $C_2^3\times D_{12}$ ($S_3$), $\GL(2,\mathbb{Z}/4):C_2$ ($S_3$)

Order 144: $C_2^2:C_6^2$ ($D_4$), $C_{12}\times A_4$ ($C_2^3$), $C_{12}\times A_4$ ($D_4$)

Order 96: $C_2^2\times D_{12}$ ($D_6$) x 2, $C_2^3\times C_{12}$ ($D_6$), $C_2^3:C_{12}$ ($D_6$), $C_4:S_4$ ($D_6$), $A_4:Q_8$ ($D_6$)

Order 72: $C_6\times A_4$ ($C_2\times D_4$)

Order 48: $C_2^3\times C_6$ ($C_3:D_4$), $C_2^2\times A_4$ ($C_3:D_4$), $C_2^2\times C_{12}$ ($C_3:D_4$), $C_2^2\times C_{12}$ ($C_2\times D_6$), $C_2\times D_{12}$ ($S_4$), $C_4\times A_4$ ($C_3:D_4$), $C_4\times A_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($D_8:C_2$)

Order 32: $C_2^3\times C_4$ ($S_3^2$)

Order 24: $D_{12}$ ($C_2\times S_4$) x 2, $C_2\times C_{12}$ ($C_2\times S_4$), $C_2^2\times C_6$ ($C_6:D_4$), $C_2\times A_4$ ($C_6:D_4$)

Order 16: $C_2^4$ ($D_6:S_3$), $C_2^2\times C_4$ ($S_3\times D_6$), $C_2^2\times C_4$ ($D_6:S_3$)

Order 12: $C_2\times C_6$ ($\GL(2,\mathbb{Z}/4)$), $C_2\times C_6$ ($D_{12}:C_2^2$), $A_4$ ($D_4:D_6$), $C_{12}$ ($C_2^2\times S_4$), $C_{12}$ ($\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($D_6:D_6$), $C_2\times C_4$ ($S_3\times S_4$)

Order 6: $C_6$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($D_6:S_4$), $C_2^2$ ($D_{12}:D_6$), $C_4$ ($D_6:S_4$), $C_4$ ($D_6\times S_4$)

Order 3: $C_3$ ($C_4.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_6:\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($C_{12}.\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $C_{12}.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 576: $C_2\times A_4\times D_{12}$ ($C_2$), $\GL(2,\mathbb{Z}/4):C_6$ ($C_2$), $C_{12}.(C_2\times S_4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $D_{12}:S_4$ ($C_2$)

Order 288: $C_2^3.C_6^2$ ($C_2^2$), $A_4\times D_{12}$ ($C_2^2$), $C_{12}:S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$)

Order 192: $C_2^3\times D_{12}$ ($S_3$), $\GL(2,\mathbb{Z}/4):C_2$ ($S_3$)

Order 144: $C_2^2:C_6^2$ ($D_4$), $C_{12}\times A_4$ ($C_2^3$), $C_{12}\times A_4$ ($D_4$)

Order 96: $C_2^3\times C_{12}$ ($D_6$), $C_2^2\times D_{12}$ ($D_6$), $C_2^3:C_{12}$ ($D_6$), $C_4:S_4$ ($D_6$), $A_4:Q_8$ ($D_6$)

Order 72: $C_6\times A_4$ ($C_2\times D_4$)

Order 48: $C_2^3\times C_6$ ($C_3:D_4$), $C_2^2\times A_4$ ($C_3:D_4$), $C_2^2\times C_{12}$ ($C_3:D_4$), $C_2^2\times C_{12}$ ($C_2\times D_6$), $C_2\times D_{12}$ ($S_4$), $C_4\times A_4$ ($C_3:D_4$), $C_4\times A_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($D_8:C_2$)

Order 32: $C_2^3\times C_4$ ($S_3^2$)

Order 24: $C_2\times C_{12}$ ($C_2\times S_4$), $D_{12}$ ($C_2\times S_4$), $C_2^2\times C_6$ ($C_6:D_4$), $C_2\times A_4$ ($C_6:D_4$)

Order 16: $C_2^4$ ($D_6:S_3$), $C_2^2\times C_4$ ($S_3\times D_6$), $C_2^2\times C_4$ ($D_6:S_3$)

Order 12: $C_2\times C_6$ ($\GL(2,\mathbb{Z}/4)$), $C_2\times C_6$ ($D_{12}:C_2^2$), $A_4$ ($D_4:D_6$), $C_{12}$ ($C_2^2\times S_4$), $C_{12}$ ($\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($D_6:D_6$), $C_2\times C_4$ ($S_3\times S_4$)

Order 6: $C_6$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($D_6:S_4$), $C_2^2$ ($D_{12}:D_6$), $C_4$ ($D_6:S_4$), $C_4$ ($D_6\times S_4$)

Order 3: $C_3$ ($C_4.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_6:\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($C_{12}.\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$C_{12}.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{12}.\GL(2,\mathbb{Z}/4)$

$\rhd$

$D_{12}:S_4$

$\rhd$

$A_4\times D_{12}$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.