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G = C248D4order 128 = 27

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

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

Aliases: C248D4, C25.35C22, C24.247C23, C23.310C24, C22.1262+ 1+4, C2.8D42, (C2×D4)⋊43D4, C22⋊C432D4, (D4×C23)⋊4C2, C221C22≀C2, C232D47C2, C243C413C2, C225(C4⋊D4), (C23×C4)⋊21C22, C23.151(C2×D4), (C22×D4)⋊4C22, C2.16(D45D4), C23.10D47C2, C2.8(C233D4), C23.8Q826C2, C23.298(C4○D4), C23.23D425C2, (C22×C4).791C23, C22.190(C22×D4), C2.C4218C22, (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), (C2×C22⋊C4)⋊12C22, (C22×C22⋊C4)⋊17C2, C22.189(C2×C4○D4), SmallGroup(128,1142)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C248D4
C1C2C22C23C24C25D4×C23 — C248D4
C1C23 — C248D4
C1C23 — C248D4
C1C23 — C248D4

Generators and relations for C248D4
 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 >

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], C243C4, C23.8Q8, C23.23D4, C23.23D4 [×2], C232D4 [×2], C23.10D4 [×2], C22×C22⋊C4, C2×C22≀C2 [×2], C2×C4⋊D4 [×2], D4×C23, C248D4
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, C233D4, D42 [×2], D45D4 [×2], C248D4

Smallest permutation representation of C248D4
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)])

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
kernelC248D4C243C4C23.8Q8C23.23D4C232D4C23.10D4C22×C22⋊C4C2×C22≀C2C2×C4⋊D4D4×C23C22⋊C4C2×D4C24C23C22
# reps111322122148442

Matrix representation of C248D4 in 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] >;

C248D4 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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