Abstract group 1248.1136: $C_{156}.D_4$

• Label: 1248.1136
• Order: \( 2^{5} \cdot 3 \cdot 13 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 13 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \cdot 13 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \cdot 13 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $28$
• Trans deg.: $624$
• Rank: $3$

Group information

Description:	$C_{156}.D_4$
Order:	\(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \)
Exponent:	\(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \)
Automorphism group:	$C_3:(C_2^5.C_2^5)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{13}$
Nilpotency class:	$2$
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	13	26	39	52	78	156
Elements	1	7	2	24	14	48	12	84	24	288	168	576	1248
Conjugacy classes	1	5	2	8	10	16	12	60	24	96	120	192	546
Divisions	1	5	1	7	5	7	1	5	1	7	5	7	52
Autjugacy classes	1	4	1	4	4	4	1	4	1	4	4	4	36

Dimension	1	2	4	8	12	24	48	96
Irr. complex chars.	312	234	0	0	0	0	0	0	546
Irr. rational chars.	8	12	5	1	8	12	5	1	52

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$624$
Rank:	$3$
Inequivalent generating triples:	$199836$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{156}=[a,b]=1, c^{a}=b^{2}c^{79}, c^{b}=c^{79} \rangle$
Permutation group:	Degree $28$ $\langle(1,2)(3,4)(5,6,8,10)(7,11,12,9), (1,3)(2,4)(5,7,8,12)(6,9,10,11), (2,4) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $C_{13}$ $\, \times\, $ $(C_2^2:Q_8)$
Semidirect product:	$(Q_8\times C_{78})$ $\,\rtimes\,$ $C_2$	$(Q_8\times C_{26})$ $\,\rtimes\,$ $C_6$	$(C_2\times C_{78})$ $\,\rtimes\,$ $Q_8$ (2)	$(C_6\times Q_8)$ $\,\rtimes\,$ $C_{26}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{156}$ . $D_4$ (2)	$C_{78}$ . $(C_2\times Q_8)$	$C_{26}$ . $(C_6\times Q_8)$	$C_6$ . $(Q_8\times C_{26})$	all 36

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

Homology

Abelianization:	$C_{2}^{2} \times C_{78} \simeq C_{2}^{3} \times C_{3} \times C_{13}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 200 subgroups in 148 conjugacy classes, 96 normal (48 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{78}$	$G/Z \simeq$ $C_2^3$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^2\times C_{78}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^2\times C_{78}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{156}.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{156}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{78}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

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

Classes of subgroups up to conjugation

Order 1248: $C_{156}.D_4$

Order 624: $C_4:C_{156}$ x 3, $C_2^2:C_{156}$ x 2, $Q_8\times C_{78}$, $C_2^2\times C_{156}$

Order 416: $C_{52}.D_4$

Order 312: $C_2\times C_{156}$ x 8, $Q_8\times C_{39}$ x 2, $C_2^2\times C_{78}$

Order 208: $C_4:C_{52}$ x 3, $C_2^2:C_{52}$ x 2, $Q_8\times C_{26}$, $C_2^2\times C_{52}$

Order 156: $C_{156}$ x 7, $C_2\times C_{78}$ x 5

Order 104: $C_2\times C_{52}$ x 8, $Q_8\times C_{13}$ x 2, $C_2^2\times C_{26}$

Order 96: $C_{12}.D_4$

Order 78: $C_{78}$ x 5

Order 52: $C_{52}$ x 7, $C_2\times C_{26}$ x 5

Order 48: $C_4:C_{12}$ x 3, $C_2^2:C_{12}$ x 2, $C_6\times Q_8$, $C_2^2\times C_{12}$

Order 39: $C_{39}$

Order 32: $C_2^2:Q_8$

Order 26: $C_{26}$ x 5

Order 24: $C_2\times C_{12}$ x 8, $C_3\times Q_8$ x 2, $C_2^2\times C_6$

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 13: $C_{13}$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 5

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

Order 6: $C_6$ x 5

Order 4: $C_4$ x 7, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1248: $C_{156}.D_4$

Order 624: $C_4:C_{156}$ x 2, $Q_8\times C_{78}$, $C_2^2\times C_{156}$, $C_2^2:C_{156}$

Order 416: $C_{52}.D_4$

Order 312: $C_2\times C_{156}$ x 5, $C_2^2\times C_{78}$, $Q_8\times C_{39}$

Order 208: $C_4:C_{52}$ x 2, $Q_8\times C_{26}$, $C_2^2\times C_{52}$, $C_2^2:C_{52}$

Order 156: $C_{156}$ x 4, $C_2\times C_{78}$ x 4

Order 104: $C_2\times C_{52}$ x 5, $C_2^2\times C_{26}$, $Q_8\times C_{13}$

Order 96: $C_{12}.D_4$

Order 78: $C_{78}$ x 4

Order 52: $C_2\times C_{26}$ x 4, $C_{52}$ x 4

Order 48: $C_4:C_{12}$ x 2, $C_6\times Q_8$, $C_2^2\times C_{12}$, $C_2^2:C_{12}$

Order 39: $C_{39}$

Order 32: $C_2^2:Q_8$

Order 26: $C_{26}$ x 4

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

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 13: $C_{13}$

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 4

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

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1248: $C_{156}.D_4$ ($C_1$)

Order 624: $C_4:C_{156}$ ($C_2$) x 3, $C_2^2:C_{156}$ ($C_2$) x 2, $Q_8\times C_{78}$ ($C_2$), $C_2^2\times C_{156}$ ($C_2$)

Order 416: $C_{52}.D_4$ ($C_3$)

Order 312: $C_2\times C_{156}$ ($C_2^2$) x 6, $C_2^2\times C_{78}$ ($C_2^2$)

Order 208: $C_4:C_{52}$ ($C_6$) x 3, $C_2^2:C_{52}$ ($C_6$) x 2, $Q_8\times C_{26}$ ($C_6$), $C_2^2\times C_{52}$ ($C_6$)

Order 156: $C_{156}$ ($D_4$) x 2, $C_2\times C_{78}$ ($Q_8$) x 2, $C_2\times C_{78}$ ($C_2^3$)

Order 104: $C_2\times C_{52}$ ($C_2\times C_6$) x 6, $C_2^2\times C_{26}$ ($C_2\times C_6$)

Order 96: $C_{12}.D_4$ ($C_{13}$)

Order 78: $C_{78}$ ($D_4:C_2$), $C_{78}$ ($C_2\times Q_8$), $C_{78}$ ($C_2\times D_4$)

Order 52: $C_2\times C_{26}$ ($C_3\times Q_8$) x 2, $C_{52}$ ($C_3\times D_4$) x 2, $C_2\times C_{26}$ ($C_2^2\times C_6$)

Order 48: $C_4:C_{12}$ ($C_{26}$) x 3, $C_2^2:C_{12}$ ($C_{26}$) x 2, $C_6\times Q_8$ ($C_{26}$), $C_2^2\times C_{12}$ ($C_{26}$)

Order 39: $C_{39}$ ($C_2^2:Q_8$)

Order 32: $C_2^2:Q_8$ ($C_{39}$)

Order 26: $C_{26}$ ($D_4:C_6$), $C_{26}$ ($C_6\times Q_8$), $C_{26}$ ($C_6\times D_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{26}$) x 6, $C_2^2\times C_6$ ($C_2\times C_{26}$)

Order 16: $C_4:C_4$ ($C_{78}$) x 3, $C_2^2:C_4$ ($C_{78}$) x 2, $C_2\times Q_8$ ($C_{78}$), $C_2^2\times C_4$ ($C_{78}$)

Order 13: $C_{13}$ ($C_{12}.D_4$)

Order 12: $C_2\times C_6$ ($Q_8\times C_{13}$) x 2, $C_{12}$ ($D_4\times C_{13}$) x 2, $C_2\times C_6$ ($C_2^2\times C_{26}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{78}$) x 6, $C_2^3$ ($C_2\times C_{78}$)

Order 6: $C_6$ ($D_4:C_{26}$), $C_6$ ($Q_8\times C_{26}$), $C_6$ ($D_4\times C_{26}$)

Order 4: $C_2^2$ ($Q_8\times C_{39}$) x 2, $C_4$ ($D_4\times C_{39}$) x 2, $C_2^2$ ($C_2^2\times C_{78}$)

Order 3: $C_3$ ($C_{52}.D_4$)

Order 2: $C_2$ ($D_4:C_{78}$), $C_2$ ($Q_8\times C_{78}$), $C_2$ ($D_4\times C_{78}$)

Order 1: $C_1$ ($C_{156}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1248: $C_{156}.D_4$ ($C_1$)

Order 624: $C_4:C_{156}$ ($C_2$) x 2, $Q_8\times C_{78}$ ($C_2$), $C_2^2\times C_{156}$ ($C_2$), $C_2^2:C_{156}$ ($C_2$)

Order 416: $C_{52}.D_4$ ($C_3$)

Order 312: $C_2\times C_{156}$ ($C_2^2$) x 4, $C_2^2\times C_{78}$ ($C_2^2$)

Order 208: $C_4:C_{52}$ ($C_6$) x 2, $Q_8\times C_{26}$ ($C_6$), $C_2^2\times C_{52}$ ($C_6$), $C_2^2:C_{52}$ ($C_6$)

Order 156: $C_{156}$ ($D_4$), $C_2\times C_{78}$ ($C_2^3$), $C_2\times C_{78}$ ($Q_8$)

Order 104: $C_2\times C_{52}$ ($C_2\times C_6$) x 4, $C_2^2\times C_{26}$ ($C_2\times C_6$)

Order 96: $C_{12}.D_4$ ($C_{13}$)

Order 78: $C_{78}$ ($D_4:C_2$), $C_{78}$ ($C_2\times Q_8$), $C_{78}$ ($C_2\times D_4$)

Order 52: $C_2\times C_{26}$ ($C_2^2\times C_6$), $C_2\times C_{26}$ ($C_3\times Q_8$), $C_{52}$ ($C_3\times D_4$)

Order 48: $C_4:C_{12}$ ($C_{26}$) x 2, $C_6\times Q_8$ ($C_{26}$), $C_2^2\times C_{12}$ ($C_{26}$), $C_2^2:C_{12}$ ($C_{26}$)

Order 39: $C_{39}$ ($C_2^2:Q_8$)

Order 32: $C_2^2:Q_8$ ($C_{39}$)

Order 26: $C_{26}$ ($D_4:C_6$), $C_{26}$ ($C_6\times Q_8$), $C_{26}$ ($C_6\times D_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{26}$) x 4, $C_2^2\times C_6$ ($C_2\times C_{26}$)

Order 16: $C_4:C_4$ ($C_{78}$) x 2, $C_2^2:C_4$ ($C_{78}$), $C_2\times Q_8$ ($C_{78}$), $C_2^2\times C_4$ ($C_{78}$)

Order 13: $C_{13}$ ($C_{12}.D_4$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_{26}$), $C_2\times C_6$ ($Q_8\times C_{13}$), $C_{12}$ ($D_4\times C_{13}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{78}$) x 4, $C_2^3$ ($C_2\times C_{78}$)

Order 6: $C_6$ ($D_4:C_{26}$), $C_6$ ($Q_8\times C_{26}$), $C_6$ ($D_4\times C_{26}$)

Order 4: $C_2^2$ ($C_2^2\times C_{78}$), $C_2^2$ ($Q_8\times C_{39}$), $C_4$ ($D_4\times C_{39}$)

Order 3: $C_3$ ($C_{52}.D_4$)

Order 2: $C_2$ ($D_4:C_{78}$), $C_2$ ($Q_8\times C_{78}$), $C_2$ ($D_4\times C_{78}$)

Order 1: $C_1$ ($C_{156}.D_4$)

Series

Derived series

$C_{156}.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{156}.D_4$

$\rhd$

$C_2^2\times C_{156}$

$\rhd$

$C_2^2\times C_{78}$

$\rhd$

$C_2\times C_{78}$

$\rhd$

$C_{78}$

$\rhd$

$C_{39}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$C_{156}.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{78}$

$\lhd$

$C_{156}.D_4$

Character theory

Complex character table

The $546 \times 546$ character table is not available for this group.

Rational character table

See the $52 \times 52$ rational character table.