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G = C2xC8:4D4order 128 = 27

Direct product of C2 and C8:4D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2xC8:4D4, C42.357D4, C42.711C23, C4:1(C2xD8), (C2xC4):7D8, (C2xC8):30D4, C8:10(C2xD4), (C4xC8):73C22, C4.2(C22xD4), (C22xD8):11C2, (C2xD8):44C22, C22.73(C2xD8), C2.11(C22xD8), C4.12(C4:1D4), C4:1D4:36C22, (C2xC4).342C24, (C2xC8).558C23, (C22xC4).612D4, C23.877(C2xD4), (C2xD4).109C23, C22.48(C4:1D4), (C22xC8).535C22, C22.602(C22xD4), (C22xC4).1557C23, (C2xC42).1126C22, (C22xD4).372C22, (C2xC4xC8):25C2, (C2xC4:1D4):17C2, (C2xC4).852(C2xD4), C2.21(C2xC4:1D4), SmallGroup(128,1876)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C2xC8:4D4
C1C2C22C2xC4C22xC4C2xC42C2xC4xC8 — C2xC8:4D4
C1C2C2xC4 — C2xC8:4D4
C1C23C2xC42 — C2xC8:4D4
C1C2C2C2xC4 — C2xC8:4D4

Generators and relations for C2xC8:4D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 980 in 380 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2xC4, C2xC4, D4, C23, C23, C42, C2xC8, D8, C22xC4, C22xC4, C2xD4, C2xD4, C24, C4xC8, C2xC42, C4:1D4, C4:1D4, C22xC8, C2xD8, C2xD8, C22xD4, C22xD4, C2xC4xC8, C8:4D4, C2xC4:1D4, C22xD8, C2xC8:4D4
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C24, C4:1D4, C2xD8, C22xD4, C8:4D4, C2xC4:1D4, C22xD8, C2xC8:4D4

Smallest permutation representation of C2xC8:4D4
On 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 32 37 59)(2 25 38 60)(3 26 39 61)(4 27 40 62)(5 28 33 63)(6 29 34 64)(7 30 35 57)(8 31 36 58)(9 43 52 20)(10 44 53 21)(11 45 54 22)(12 46 55 23)(13 47 56 24)(14 48 49 17)(15 41 50 18)(16 42 51 19)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 56)(41 60)(42 59)(43 58)(44 57)(45 64)(46 63)(47 62)(48 61)

G:=sub<Sym(64)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,32,37,59)(2,25,38,60)(3,26,39,61)(4,27,40,62)(5,28,33,63)(6,29,34,64)(7,30,35,57)(8,31,36,58)(9,43,52,20)(10,44,53,21)(11,45,54,22)(12,46,55,23)(13,47,56,24)(14,48,49,17)(15,41,50,18)(16,42,51,19), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,56)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61)>;

G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,32,37,59)(2,25,38,60)(3,26,39,61)(4,27,40,62)(5,28,33,63)(6,29,34,64)(7,30,35,57)(8,31,36,58)(9,43,52,20)(10,44,53,21)(11,45,54,22)(12,46,55,23)(13,47,56,24)(14,48,49,17)(15,41,50,18)(16,42,51,19), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,56)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61) );

G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,32,37,59),(2,25,38,60),(3,26,39,61),(4,27,40,62),(5,28,33,63),(6,29,34,64),(7,30,35,57),(8,31,36,58),(9,43,52,20),(10,44,53,21),(11,45,54,22),(12,46,55,23),(13,47,56,24),(14,48,49,17),(15,41,50,18),(16,42,51,19)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,26),(18,25),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,56),(41,60),(42,59),(43,58),(44,57),(45,64),(46,63),(47,62),(48,61)]])

44 conjugacy classes

class 1 2A···2G2H···2O4A···4L8A···8P
order12···22···24···48···8
size11···18···82···22···2

44 irreducible representations

dim111112222
type+++++++++
imageC1C2C2C2C2D4D4D4D8
kernelC2xC8:4D4C2xC4xC8C8:4D4C2xC4:1D4C22xD8C42C2xC8C22xC4C2xC4
# reps1182428216

Matrix representation of C2xC8:4D4 in GL5(F17)

160000
01000
00100
000160
000016
,
160000
001100
031100
000143
0001414
,
160000
011500
011600
00010
00001
,
10000
01000
011600
00010
000016

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,3,0,0,0,11,11,0,0,0,0,0,14,14,0,0,0,3,14],[16,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,1,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16] >;

C2xC8:4D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_4D_4
% in TeX

G:=Group("C2xC8:4D4");
// GroupNames label

G:=SmallGroup(128,1876);
// by ID

G=gap.SmallGroup(128,1876);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,184,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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