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G = C2×C87D4order 128 = 27

Direct product of C2 and C87D4

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

Aliases: C2×C87D4, C234D8, C24.142D4, C816(C2×D4), (C2×C8)⋊37D4, (C23×C8)⋊8C2, C221(C2×D8), C2.6(C22×D8), (C22×D8)⋊10C2, (C2×D8)⋊42C22, C4⋊C4.17C23, C2.D844C22, C4⋊D453C22, (C2×C8).486C23, (C2×C4).252C24, (C22×C8)⋊65C22, (C2×D4).57C23, C23.859(C2×D4), (C22×C4).606D4, C4.146(C22×D4), C4.108(C4⋊D4), D4⋊C457C22, C22.91(C4○D8), (C23×C4).701C22, C22.512(C22×D4), C22.172(C4⋊D4), (C22×C4).1531C23, (C22×D4).345C22, (C2×C2.D8)⋊18C2, C2.14(C2×C4○D8), C4.19(C2×C4○D4), (C2×C4⋊D4)⋊46C2, C2.70(C2×C4⋊D4), (C2×D4⋊C4)⋊17C2, (C2×C4).1424(C2×D4), (C2×C4).698(C4○D4), (C2×C4⋊C4).586C22, SmallGroup(128,1780)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C87D4
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C2×C87D4
C1C2C2×C4 — C2×C87D4
C1C23C23×C4 — C2×C87D4
C1C2C2C2×C4 — C2×C87D4

Generators and relations for C2×C87D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 692 in 308 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×28], C23, C23 [×6], C23 [×20], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×10], D8 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×26], C24, C24 [×2], D4⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×D8 [×4], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C2×D4⋊C4 [×2], C2×C2.D8, C87D4 [×8], C2×C4⋊D4 [×2], C23×C8, C22×D8, C2×C87D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D8 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×D8 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C87D4 [×4], C2×C4⋊D4, C22×D8, C2×C4○D8, C2×C87D4

Smallest permutation representation of C2×C87D4
On 64 points
Generators in S64
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(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)
(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 42 9 19)(2 41 10 18)(3 48 11 17)(4 47 12 24)(5 46 13 23)(6 45 14 22)(7 44 15 21)(8 43 16 20)(25 50 60 34)(26 49 61 33)(27 56 62 40)(28 55 63 39)(29 54 64 38)(30 53 57 37)(31 52 58 36)(32 51 59 35)
(1 51)(2 50)(3 49)(4 56)(5 55)(6 54)(7 53)(8 52)(9 35)(10 34)(11 33)(12 40)(13 39)(14 38)(15 37)(16 36)(17 61)(18 60)(19 59)(20 58)(21 57)(22 64)(23 63)(24 62)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L8A···8P
order12···2222222224···444448···8
size11···1222288882···288882···2

44 irreducible representations

dim1111111222222
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C4○D4D8C4○D8
kernelC2×C87D4C2×D4⋊C4C2×C2.D8C87D4C2×C4⋊D4C23×C8C22×D8C2×C8C22×C4C24C2×C4C23C22
# reps1218211431488

Matrix representation of C2×C87D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1430000
14140000
0014300
00141400
0000160
0000016
,
1600000
010000
0016000
000100
0000162
0000161
,
100000
0160000
0016000
000100
000010
0000116

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

C2×C87D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_7D_4
% in TeX

G:=Group("C2xC8:7D4");
// GroupNames label

G:=SmallGroup(128,1780);
// by ID

G=gap.SmallGroup(128,1780);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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