Abstract group $C_{68}.S_4$

• Label: 1632.1025
• Order: \( 2^{5} \cdot 3 \cdot 17 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{10} \cdot 3 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $29$
• Trans deg.: $408$
• Rank: $2$

Group information

Description:	$C_{68}.S_4$
Order:	\(1632\)\(\medspace = 2^{5} \cdot 3 \cdot 17 \)
Exponent:	\(204\)\(\medspace = 2^{2} \cdot 3 \cdot 17 \)
Automorphism group:	Group of order \(52224\)\(\medspace = 2^{10} \cdot 3 \cdot 17 \) (generators)
Outer automorphisms:	$C_2^2\times C_{16}$, of order \(64\)\(\medspace = 2^{6} \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{17}$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12	17	34	51	68	102	204
Elements	1	7	8	824	8	16	16	112	128	128	128	256	1632
Conjugacy classes	1	3	1	6	1	2	8	24	16	32	16	32	142
Divisions	1	3	1	6	1	1	1	3	1	2	1	1	22
Autjugacy classes	1	3	1	4	1	1	1	3	1	2	1	1	20

Dimension	1	2	3	4	6	16	32	48	64	96
Irr. complex chars.	4	101	4	0	33	0	0	0	0	0	142
Irr. rational chars.	4	3	4	1	1	2	3	2	1	1	22

Minimal Presentations

Permutation degree:	$29$
Transitive degree:	$408$
Rank:	$2$
Inequivalent generating pairs:	$9$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	192
Arbitrary	5	7	23

Constructions

Presentation:	$\langle a, b, c, d \mid b^{12}=c^{2}=d^{34}=[a,c]=[c,d]=1, a^{2}=b^{6}, b^{a}=b^{11}, d^{a}=cd^{33}, c^{b}=cd^{17}, d^{b}=cd^{18} \rangle$
Permutation group:	Degree $29$ $\langle(2,4)(5,6,8,10)(7,11,12,9)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(A_4\times C_{17})$ $\,\rtimes\,$ $Q_8$	$C_{17}$ $\,\rtimes\,$ $(A_4:Q_8)$	$A_4$ $\,\rtimes\,$ $(C_{17}:Q_8)$	$C_2^2$ $\,\rtimes\,$ $(C_{51}:Q_8)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{68}$ . $S_4$	$C_2^3$ . $D_{102}$	$(C_{34}.S_4)$ . $C_2$ (2)	$C_{34}$ . $(C_2\times S_4)$	all 13

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

Homology

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

Subgroups

There are 1596 subgroups in 84 conjugacy classes, 21 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{34}:S_4$
Commutator:	$G' \simeq$ $A_4\times C_{34}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{34}:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{68}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_{68}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{34}$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

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 1632: $C_{68}.S_4$

Order 816: $C_{34}.S_4$ x 2, $A_4\times C_{68}$

Order 544: $C_{68}.D_4$

Order 408: $A_4\times C_{34}$, $C_{51}:Q_8$

Order 272: $C_{34}.D_4$ x 2, $C_{34}.D_4$ x 2, $C_2^2\times C_{68}$, $C_{34}:Q_8$, $C_{68}:C_4$

Order 204: $C_{51}:C_4$ x 2, $A_4\times C_{17}$, $C_{204}$

Order 136: $C_{34}:C_4$ x 4, $C_2\times C_{68}$ x 2, $C_{17}:Q_8$ x 2, $C_2^2\times C_{34}$

Order 102: $C_{102}$

Order 96: $A_4:Q_8$

Order 68: $C_{17}:C_4$ x 4, $C_2\times C_{34}$ x 3, $C_{68}$ x 2

Order 51: $C_{51}$

Order 48: $A_4:C_4$ x 2, $C_4\times A_4$

Order 34: $C_{34}$ x 3

Order 32: $C_2^2:Q_8$

Order 24: $C_3:Q_8$, $C_2\times A_4$

Order 17: $C_{17}$

Order 16: $C_4:C_4$ x 3, $C_2^2:C_4$ x 2, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $C_3:C_4$ x 2, $A_4$, $C_{12}$

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 2, $C_2^3$

Order 6: $C_6$

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1632: $C_{68}.S_4$

Order 816: $A_4\times C_{68}$, $C_{34}.S_4$

Order 544: $C_{68}.D_4$

Order 408: $A_4\times C_{34}$, $C_{51}:Q_8$

Order 272: $C_2^2\times C_{68}$, $C_{34}:Q_8$, $C_{34}.D_4$, $C_{68}:C_4$, $C_{34}.D_4$

Order 204: $A_4\times C_{17}$, $C_{204}$, $C_{51}:C_4$

Order 136: $C_2\times C_{68}$ x 2, $C_{34}:C_4$ x 2, $C_{17}:Q_8$ x 2, $C_2^2\times C_{34}$

Order 102: $C_{102}$

Order 96: $A_4:Q_8$

Order 68: $C_2\times C_{34}$ x 3, $C_{68}$ x 2, $C_{17}:C_4$ x 2

Order 51: $C_{51}$

Order 48: $C_4\times A_4$, $A_4:C_4$

Order 34: $C_{34}$ x 3

Order 32: $C_2^2:Q_8$

Order 24: $C_3:Q_8$, $C_2\times A_4$

Order 17: $C_{17}$

Order 16: $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $A_4$, $C_{12}$, $C_3:C_4$

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

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1632: $C_{68}.S_4$ ($C_1$)

Order 816: $C_{34}.S_4$ ($C_2$) x 2, $A_4\times C_{68}$ ($C_2$)

Order 408: $A_4\times C_{34}$ ($C_2^2$)

Order 272: $C_2^2\times C_{68}$ ($S_3$)

Order 204: $A_4\times C_{17}$ ($Q_8$)

Order 136: $C_2^2\times C_{34}$ ($D_6$)

Order 68: $C_2\times C_{34}$ ($C_3:Q_8$), $C_{68}$ ($S_4$)

Order 48: $C_4\times A_4$ ($D_{17}$)

Order 34: $C_{34}$ ($C_2\times S_4$)

Order 24: $C_2\times A_4$ ($D_{34}$)

Order 17: $C_{17}$ ($A_4:Q_8$)

Order 16: $C_2^2\times C_4$ ($D_{51}$)

Order 12: $A_4$ ($C_{17}:Q_8$)

Order 8: $C_2^3$ ($D_{102}$)

Order 4: $C_2^2$ ($C_{51}:Q_8$), $C_4$ ($C_{17}:S_4$)

Order 2: $C_2$ ($C_{34}:S_4$)

Order 1: $C_1$ ($C_{68}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1632: $C_{68}.S_4$ ($C_1$)

Order 816: $A_4\times C_{68}$ ($C_2$), $C_{34}.S_4$ ($C_2$)

Order 408: $A_4\times C_{34}$ ($C_2^2$)

Order 272: $C_2^2\times C_{68}$ ($S_3$)

Order 204: $A_4\times C_{17}$ ($Q_8$)

Order 136: $C_2^2\times C_{34}$ ($D_6$)

Order 68: $C_2\times C_{34}$ ($C_3:Q_8$), $C_{68}$ ($S_4$)

Order 48: $C_4\times A_4$ ($D_{17}$)

Order 34: $C_{34}$ ($C_2\times S_4$)

Order 24: $C_2\times A_4$ ($D_{34}$)

Order 17: $C_{17}$ ($A_4:Q_8$)

Order 16: $C_2^2\times C_4$ ($D_{51}$)

Order 12: $A_4$ ($C_{17}:Q_8$)

Order 8: $C_2^3$ ($D_{102}$)

Order 4: $C_2^2$ ($C_{51}:Q_8$), $C_4$ ($C_{17}:S_4$)

Order 2: $C_2$ ($C_{34}:S_4$)

Order 1: $C_1$ ($C_{68}.S_4$)

Series

Derived series

$C_{68}.S_4$

$\rhd$

$A_4\times C_{34}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{68}.S_4$

$\rhd$

$C_{34}.S_4$

$\rhd$

$A_4\times C_{34}$

$\rhd$

$A_4\times C_{17}$

$\rhd$

$C_2\times C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{68}.S_4$

$\rhd$

$A_4\times C_{34}$

$\rhd$

$A_4\times C_{17}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Character theory

Complex character table

See the $142 \times 142$ 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 $22 \times 22$ rational character table.