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G = C24⋊8D4  order 128 = 27

3rd semidirect product of C24 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24⋊8D4, C25.35C22, C24.247C23, C23.310C24, C22.1262+ (1+4), C2.8(D42), (C2×D4)⋊43D4, C22⋊C4⋊32D4, (D4×C23)⋊4C2, C22⋊1C22≀C2, C23⋊2D4⋊7C2, C24⋊3C4⋊13C2, C22⋊5(C4⋊D4), (C23×C4)⋊21C22, C23.151(C2×D4), (C22×D4)⋊4C22, C2.16(D4⋊5D4), C23.10D4⋊7C2, C2.8(C23⋊3D4), C23.8Q8⋊26C2, C23.298(C4○D4), C23.23D4⋊25C2, (C22×C4).791C23, C22.190(C22×D4), C2.C42⋊18C22, (C2×C4)⋊9(C2×D4), (C2×C4⋊D4)⋊4C2, (C2×C22≀C2)⋊5C2, (C2×C4⋊C4)⋊11C22, C2.14(C2×C4⋊D4), C2.17(C2×C22≀C2), (C22×C22⋊C4)⋊17C2, (C2×C22⋊C4)⋊12C22, C22.189(C2×C4○D4), SmallGroup(128,1142)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C24⋊8D4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C24⋊8D4
Lower central
C1 — C23 — C24⋊8D4
Upper central
C1 — C23 — C24⋊8D4
Jennings
C1 — C23 — C24⋊8D4

Subgroups: 1412 in 623 conjugacy classes, 132 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×15], C4 [×13], C22 [×3], C22 [×12], C22 [×93], C2×C4 [×8], C2×C4 [×39], D4 [×52], C23, C23 [×18], C23 [×101], C22⋊C4 [×4], C22⋊C4 [×26], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×6], C22×C4 [×16], C2×D4 [×8], C2×D4 [×78], C24 [×4], C24 [×6], C24 [×22], C2.C42 [×4], C2×C22⋊C4 [×5], C2×C22⋊C4 [×10], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C22≀C2 [×8], C4⋊D4 [×8], C23×C4, C23×C4 [×2], C22×D4 [×3], C22×D4 [×6], C22×D4 [×12], C25 [×2], C24⋊3C4, C23.8Q8, C23.23D4, C23.23D4 [×2], C23⋊2D4 [×2], C23.10D4 [×2], C22×C22⋊C4, C2×C22≀C2 [×2], C2×C4⋊D4 [×2], D4×C23, C24⋊8D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×16], C23 [×15], C2×D4 [×24], C4○D4 [×2], C24, C22≀C2 [×4], C4⋊D4 [×4], C22×D4 [×4], C2×C4○D4, 2+ (1+4) [×2], C2×C22≀C2, C2×C4⋊D4, C23⋊3D4, D42 [×2], D4⋊5D4 [×2], C24⋊8D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Smallest permutation representation
►On 32 points
Generators in S32
(1 11)(2 8)(3 9)(4 6)(5 21)(7 23)(10 22)(12 24)(13 20)(14 30)(15 18)(16 32)(17 26)(19 28)(25 29)(27 31)

(1 25)(2 14)(3 27)(4 16)(5 18)(6 32)(7 20)(8 30)(9 31)(10 19)(11 29)(12 17)(13 23)(15 21)(22 28)(24 26)

(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)

(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

(1 2)(3 4)(5 17)(6 20)(7 19)(8 18)(9 30)(10 29)(11 32)(12 31)(13 14)(15 16)(21 22)(23 24)(25 26)(27 28)
 
G:=sub<Sym(32)| (1,11)(2,8)(3,9)(4,6)(5,21)(7,23)(10,22)(12,24)(13,20)(14,30)(15,18)(16,32)(17,26)(19,28)(25,29)(27,31), (1,25)(2,14)(3,27)(4,16)(5,18)(6,32)(7,20)(8,30)(9,31)(10,19)(11,29)(12,17)(13,23)(15,21)(22,28)(24,26), (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,2)(3,4)(5,17)(6,20)(7,19)(8,18)(9,30)(10,29)(11,32)(12,31)(13,14)(15,16)(21,22)(23,24)(25,26)(27,28)>;
 
G:=Group( (1,11)(2,8)(3,9)(4,6)(5,21)(7,23)(10,22)(12,24)(13,20)(14,30)(15,18)(16,32)(17,26)(19,28)(25,29)(27,31), (1,25)(2,14)(3,27)(4,16)(5,18)(6,32)(7,20)(8,30)(9,31)(10,19)(11,29)(12,17)(13,23)(15,21)(22,28)(24,26), (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,2)(3,4)(5,17)(6,20)(7,19)(8,18)(9,30)(10,29)(11,32)(12,31)(13,14)(15,16)(21,22)(23,24)(25,26)(27,28) );
 
G=PermutationGroup([(1,11),(2,8),(3,9),(4,6),(5,21),(7,23),(10,22),(12,24),(13,20),(14,30),(15,18),(16,32),(17,26),(19,28),(25,29),(27,31)], [(1,25),(2,14),(3,27),(4,16),(5,18),(6,32),(7,20),(8,30),(9,31),(10,19),(11,29),(12,17),(13,23),(15,21),(22,28),(24,26)], [(1,23),(2,24),(3,21),(4,22),(5,9),(6,10),(7,11),(8,12),(13,25),(14,26),(15,27),(16,28),(17,30),(18,31),(19,32),(20,29)], [(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,2),(3,4),(5,17),(6,20),(7,19),(8,18),(9,30),(10,29),(11,32),(12,31),(13,14),(15,16),(21,22),(23,24),(25,26),(27,28)])
 

Matrix representation ►G ⊆ GL6(ℤ)

100000
010000
001000
002-100
00001-2
00000-1
,
-100000
0-10000
001000
002-100
0000-10
00000-1
,
100000
010000
00-1000
000-100
000010
000001
,
100000
010000
001000
000100
0000-10
00000-1
,
1-20000
1-10000
001-100
000-100
0000-10
00000-1
,
-120000
010000
001-100
000-100
0000-10
0000-11

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,2,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A···2G2H···2O2P···2U2V4A···4L4M4N4O
order12···22···22···224···4444
size11···12···24···484···4888

38 irreducible representations

dim111111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D42+ (1+4)
kernelC24⋊8D4C24⋊3C4C23.8Q8C23.23D4C23⋊2D4C23.10D4C22×C22⋊C4C2×C22≀C2C2×C4⋊D4D4×C23C22⋊C4C2×D4C24C23C22
# reps111322122148442

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_8D_4
 
% in TeX
 
G:=Group("C2^4:8D4");
 
// GroupNames label
 
G:=SmallGroup(128,1142);
 
// by ID
 
G=gap.SmallGroup(128,1142);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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