Abstract group 24704.21819: $C_{386}:C_{64}$

• Label: 24704.21819
• Order: \( 2^{7} \cdot 193 \)
• Exponent: \( 2^{6} \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $195$
• Trans deg.: $386$
• Rank: $2$

Group information

Description:	$C_{386}:C_{64}$
Order:	\(24704\)\(\medspace = 2^{7} \cdot 193 \)
Exponent:	\(12352\)\(\medspace = 2^{6} \cdot 193 \)
Automorphism group:	$C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 7, $C_{193}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	4	8	16	32	64	193	386
Elements	1	387	772	1544	3088	6176	12352	192	192	24704
Conjugacy classes	1	3	4	8	16	32	64	3	3	134
Divisions	1	3	2	2	2	2	2	1	1	16
Autjugacy classes	1	3	4	8	16	32	32	1	1	98

Dimension	1	2	4	8	16	32	64	192
Irr. complex chars.	128	0	0	0	0	0	6	0	134
Irr. rational chars.	4	2	2	2	2	2	0	2	16

Minimal presentations

Permutation degree:	$195$
Transitive degree:	$386$
Rank:	$2$
Inequivalent generating pairs:	$3072$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	64	64	192
Arbitrary	64	64	192

Constructions

Presentation:	$\langle a, b \mid a^{64}=b^{386}=1, b^{a}=b^{105} \rangle$
Permutation group:	Degree $195$ $\langle(2,4,5,28,6,29,35,50,7,30,36,68,41,67,52,10,8,27,37,44,42,90,96,49,25,85,87,95,24,84,23,22,3,17,18,83,19,69,31,91,20,79,86,93,75,77,74,21,15,16,34,40,45,47,73,71,14,33,39,70,13,32,12,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 94 & 0 \\ 0 & 143 \end{array}\right), \left(\begin{array}{rr} 192 & 0 \\ 0 & 192 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_2$ $\, \times\, $ $(C_{193}:C_{64})$
Semidirect product:	$D_{193}$ $\,\rtimes\,$ $C_{64}$	$C_{386}$ $\,\rtimes\,$ $C_{64}$	$C_{193}$ $\,\rtimes\,$ $(C_2\times C_{64})$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{386}$ . $C_{32}$	$(C_{386}:C_8)$ . $C_8$	$D_{193}$ . $(C_2\times C_{32})$	$(C_{386}:C_{32})$ . $C_2$	all 14

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

Homology

Abelianization:	$C_{2} \times C_{64} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 3496 subgroups in 40 conjugacy classes, 22 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{193}:C_{64}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_2\times C_{64}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{386}:C_{64}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{386}$	$G/\operatorname{Fit} \simeq$ $C_{64}$
Radical:	$R \simeq$ $C_{386}:C_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_{64}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{64}$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

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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Classes of subgroups up to conjugation

Order 24704: $C_{386}:C_{64}$

Order 12352: $C_{193}:C_{64}$ x 2, $C_{386}:C_{32}$

Order 6176: $C_{193}:C_{32}$ x 2, $C_{386}:C_{16}$

Order 3088: $C_{193}:C_{16}$ x 2, $C_{386}:C_8$

Order 1544: $C_{193}:C_8$ x 2, $C_{386}:C_4$

Order 772: $C_{193}:C_4$ x 2, $D_{386}$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 193: $C_{193}$

Order 128: $C_2\times C_{64}$

Order 64: $C_{64}$ x 2, $C_2\times C_{32}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 24704: $C_{386}:C_{64}$

Order 12352: $C_{193}:C_{64}$, $C_{386}:C_{32}$

Order 6176: $C_{193}:C_{32}$ x 2, $C_{386}:C_{16}$

Order 3088: $C_{193}:C_{16}$ x 2, $C_{386}:C_8$

Order 1544: $C_{193}:C_8$ x 2, $C_{386}:C_4$

Order 772: $C_{193}:C_4$ x 2, $D_{386}$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 193: $C_{193}$

Order 128: $C_2\times C_{64}$

Order 64: $C_2\times C_{32}$, $C_{64}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24704: $C_{386}:C_{64}$ ($C_1$)

Order 12352: $C_{193}:C_{64}$ ($C_2$) x 2, $C_{386}:C_{32}$ ($C_2$)

Order 6176: $C_{193}:C_{32}$ ($C_2^2$), $C_{193}:C_{32}$ ($C_4$), $C_{386}:C_{16}$ ($C_4$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_4$), $C_{193}:C_{16}$ ($C_8$), $C_{386}:C_8$ ($C_8$)

Order 1544: $C_{193}:C_8$ ($C_2\times C_8$), $C_{193}:C_8$ ($C_{16}$), $C_{386}:C_4$ ($C_{16}$)

Order 772: $D_{386}$ ($C_{32}$), $C_{193}:C_4$ ($C_2\times C_{16}$), $C_{193}:C_4$ ($C_{32}$)

Order 386: $C_{386}$ ($C_{64}$), $D_{193}$ ($C_2\times C_{32}$), $D_{193}$ ($C_{64}$)

Order 193: $C_{193}$ ($C_2\times C_{64}$)

Order 2: $C_2$ ($C_{193}:C_{64}$)

Order 1: $C_1$ ($C_{386}:C_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24704: $C_{386}:C_{64}$ ($C_1$)

Order 12352: $C_{193}:C_{64}$ ($C_2$), $C_{386}:C_{32}$ ($C_2$)

Order 6176: $C_{193}:C_{32}$ ($C_2^2$), $C_{193}:C_{32}$ ($C_4$), $C_{386}:C_{16}$ ($C_4$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_4$), $C_{193}:C_{16}$ ($C_8$), $C_{386}:C_8$ ($C_8$)

Order 1544: $C_{193}:C_8$ ($C_2\times C_8$), $C_{193}:C_8$ ($C_{16}$), $C_{386}:C_4$ ($C_{16}$)

Order 772: $D_{386}$ ($C_{32}$), $C_{193}:C_4$ ($C_2\times C_{16}$), $C_{193}:C_4$ ($C_{32}$)

Order 386: $C_{386}$ ($C_{64}$), $D_{193}$ ($C_2\times C_{32}$), $D_{193}$ ($C_{64}$)

Order 193: $C_{193}$ ($C_2\times C_{64}$)

Order 2: $C_2$ ($C_{193}:C_{64}$)

Order 1: $C_1$ ($C_{386}:C_{64}$)

Series

Derived series

$C_{386}:C_{64}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{386}:C_{64}$

$\rhd$

$C_{193}:C_{64}$

$\rhd$

$C_{193}:C_{32}$

$\rhd$

$C_{193}:C_{16}$

$\rhd$

$C_{193}:C_8$

$\rhd$

$C_{193}:C_4$

$\rhd$

$D_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{386}:C_{64}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 3 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 $134 \times 134$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	8A	8B	16A	16B	32A	32B	64A	64B	193A	386A
	Size	1	1	193	193	386	386	772	772	1544	1544	3088	3088	6176	6176	192	192
	2 P	1A	1A	1A	1A	2B	2B	4A	4A	8A	8A	16B	16B	32A	32A	193A	193A
	193 P	1A	2A	2B	2C	4A	4B	8A	8B	16A	16B	32A	32B	64A	64B	193A	386A
24704.21819.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24704.21819.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$
24704.21819.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$
24704.21819.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
24704.21819.1e		$2$	$-2$	$2$	$-2$	$2$	$-2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$-2$
24704.21819.1f		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$2$
24704.21819.1g		$4$	$-4$	$4$	$-4$	$4$	$-4$	$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$	$4$	$-4$
24704.21819.1h		$4$	$4$	$4$	$4$	$4$	$4$	$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$	$4$	$4$
24704.21819.1i		$8$	$-8$	$8$	$-8$	$8$	$-8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$8$	$-8$
24704.21819.1j		$8$	$8$	$8$	$8$	$8$	$8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$8$	$8$
24704.21819.1k		$16$	$-16$	$16$	$-16$	$-16$	$16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$16$	$-16$
24704.21819.1l		$16$	$16$	$16$	$16$	$-16$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$16$	$16$
24704.21819.1m		$32$	$-32$	$-32$	$32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$32$	$-32$
24704.21819.1n		$32$	$32$	$-32$	$-32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$32$	$32$
24704.21819.64a		$192$	$192$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
24704.21819.64b		$192$	$-192$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$