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G = C2×C84D4order 128 = 27

Direct product of C2 and C84D4

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

Aliases: C2×C84D4, C42.357D4, C42.711C23, C41(C2×D8), (C2×C4)⋊7D8, (C2×C8)⋊30D4, C810(C2×D4), (C4×C8)⋊73C22, C4.2(C22×D4), (C22×D8)⋊11C2, (C2×D8)⋊44C22, C22.73(C2×D8), C2.11(C22×D8), C4.12(C41D4), C41D436C22, (C2×C4).342C24, (C2×C8).558C23, (C22×C4).612D4, C23.877(C2×D4), (C2×D4).109C23, C22.48(C41D4), (C22×C8).535C22, C22.602(C22×D4), (C22×C4).1557C23, (C2×C42).1126C22, (C22×D4).372C22, (C2×C4×C8)⋊25C2, (C2×C41D4)⋊17C2, (C2×C4).852(C2×D4), C2.21(C2×C41D4), SmallGroup(128,1876)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C84D4
C1C2C22C2×C4C22×C4C2×C42C2×C4×C8 — C2×C84D4
C1C2C2×C4 — C2×C84D4
C1C23C2×C42 — C2×C84D4
C1C2C2C2×C4 — C2×C84D4

Generators and relations for C2×C84D4
 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 [×6], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×40], C8 [×8], C2×C4 [×2], C2×C4 [×16], D4 [×56], C23, C23 [×32], C42 [×4], C2×C8 [×12], D8 [×32], C22×C4, C22×C4 [×2], C2×D4 [×8], C2×D4 [×52], C24 [×4], C4×C8 [×4], C2×C42, C41D4 [×8], C41D4 [×4], C22×C8 [×2], C2×D8 [×16], C2×D8 [×16], C22×D4 [×4], C22×D4 [×4], C2×C4×C8, C84D4 [×8], C2×C41D4 [×2], C22×D8 [×4], C2×C84D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D8 [×8], C2×D4 [×18], C24, C41D4 [×4], C2×D8 [×12], C22×D4 [×3], C84D4 [×4], C2×C41D4, C22×D8 [×2], C2×C84D4

Smallest permutation representation of C2×C84D4
On 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 39 59)(2 25 40 60)(3 26 33 61)(4 27 34 62)(5 28 35 63)(6 29 36 64)(7 30 37 57)(8 31 38 58)(9 43 52 24)(10 44 53 17)(11 45 54 18)(12 46 55 19)(13 47 56 20)(14 48 49 21)(15 41 50 22)(16 42 51 23)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)(33 49)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(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)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,32,39,59)(2,25,40,60)(3,26,33,61)(4,27,34,62)(5,28,35,63)(6,29,36,64)(7,30,37,57)(8,31,38,58)(9,43,52,24)(10,44,53,17)(11,45,54,18)(12,46,55,19)(13,47,56,20)(14,48,49,21)(15,41,50,22)(16,42,51,23), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31)(33,49)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(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)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,32,39,59)(2,25,40,60)(3,26,33,61)(4,27,34,62)(5,28,35,63)(6,29,36,64)(7,30,37,57)(8,31,38,58)(9,43,52,24)(10,44,53,17)(11,45,54,18)(12,46,55,19)(13,47,56,20)(14,48,49,21)(15,41,50,22)(16,42,51,23), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31)(33,49)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(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),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)], [(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)], [(1,32,39,59),(2,25,40,60),(3,26,33,61),(4,27,34,62),(5,28,35,63),(6,29,36,64),(7,30,37,57),(8,31,38,58),(9,43,52,24),(10,44,53,17),(11,45,54,18),(12,46,55,19),(13,47,56,20),(14,48,49,21),(15,41,50,22),(16,42,51,23)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,32),(24,31),(33,49),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(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
kernelC2×C84D4C2×C4×C8C84D4C2×C41D4C22×D8C42C2×C8C22×C4C2×C4
# reps1182428216

Matrix representation of C2×C84D4 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] >;

C2×C84D4 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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