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G = C2xC8:3D4order 128 = 27

Direct product of C2 and C8:3D4

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

Aliases: C2xC8:3D4, C42.245D4, C42.370C23, C8:7(C2xD4), (C2xC8):15D4, C4.6(C22xD4), (C22xD8):18C2, (C2xD8):50C22, C8:C4:44C22, C4.15(C4:1D4), C4:1D4:37C22, (C2xC8).264C23, (C2xC4).346C24, (C22xSD16):4C2, (C22xC4).465D4, C23.880(C2xD4), (C2xSD16):58C22, (C2xD4).112C23, C4.4D4:57C22, (C2xQ8).100C23, C22.51(C4:1D4), (C22xC8).268C22, (C2xC42).852C22, C22.606(C22xD4), C22.125(C8:C22), (C22xC4).1561C23, (C22xD4).374C22, (C22xQ8).307C22, (C2xC8:C4):8C2, (C2xC4:1D4):18C2, (C2xC4).856(C2xD4), C2.25(C2xC4:1D4), C2.41(C2xC8:C22), (C2xC4.4D4):41C2, SmallGroup(128,1880)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C2xC8:3D4
C1C2C22C2xC4C22xC4C2xC42C2xC8:C4 — C2xC8:3D4
C1C2C2xC4 — C2xC8:3D4
C1C23C2xC42 — C2xC8:3D4
C1C2C2C2xC4 — C2xC8:3D4

Generators and relations for C2xC8:3D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 804 in 334 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C2xC8, D8, SD16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C8:C4, C2xC42, C2xC22:C4, C4.4D4, C4.4D4, C4:1D4, C4:1D4, C22xC8, C2xD8, C2xD8, C2xSD16, C2xSD16, C22xD4, C22xD4, C22xD4, C22xQ8, C2xC8:C4, C8:3D4, C2xC4.4D4, C2xC4:1D4, C22xD8, C22xSD16, C2xC8:3D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4:1D4, C8:C22, C22xD4, C8:3D4, C2xC4:1D4, C2xC8:C22, C2xC8:3D4

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12···22···244444444448···8
size11···18···822224444884···4

32 irreducible representations

dim11111112224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C8:C22
kernelC2xC8:3D4C2xC8:C4C8:3D4C2xC4.4D4C2xC4:1D4C22xD8C22xSD16C42C2xC8C22xC4C22
# reps11811222824

Matrix representation of C2xC8:3D4 in GL8(F17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
161000000
151000000
000160000
00100000
00001315011
00001030
000013602
0000513164
,
116000000
216000000
000160000
00100000
000000160
00001601616
000016000
000011610
,
10000000
216000000
001600000
00010000
000016000
000016100
000000160
00002011

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,15,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,1,13,5,0,0,0,0,15,0,6,13,0,0,0,0,0,3,0,16,0,0,0,0,11,0,2,4],[1,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,0,16,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0],[1,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1] >;

C2xC8:3D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_3D_4
% in TeX

G:=Group("C2xC8:3D4");
// GroupNames label

G:=SmallGroup(128,1880);
// by ID

G=gap.SmallGroup(128,1880);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,184,2804,172]);
// 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,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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