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G = C22×C8○D4order 128 = 27

Direct product of C22 and C8○D4

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

Aliases: C22×C8○D4, C8.21C24, C4.20C25, M4(2)⋊13C23, D4(C22×C8), C8(C22×D4), Q8(C22×C8), C8(C22×Q8), (C23×C8)⋊17C2, (C2×C8)⋊18C23, C4.66(C23×C4), C2.14(C24×C4), C24.102(C2×C4), (C2×C4).604C24, (C22×C8)⋊72C22, C4○D4.35C23, (C22×D4).47C4, D4.26(C22×C4), M4(2)2(C22×C8), C82(C22×M4(2)), C22.7(C23×C4), (C22×Q8).37C4, Q8.27(C22×C4), (C2×M4(2))⋊82C22, (C22×M4(2))⋊29C2, (C23×C4).710C22, C23.156(C22×C4), (C22×C4).1588C23, C4(C2×C8○D4), C82(C2×C4○D4), C82(C2×C8○D4), (C2×C8)2(C2×D4), (C2×C8)2(C2×Q8), (C2×C4)(C8○D4), C8(C22×C4○D4), C4○D4(C22×C8), (C2×C8)2(C8○D4), (C2×C8)2(C4○D4), (C2×C8)(C22×Q8), (C2×Q8)(C22×C8), C4○D4.40(C2×C4), (C2×C4○D4).36C4, (C2×C8)3(C2×M4(2)), (C2×D4).254(C2×C4), (C22×C8)(C22×Q8), (C2×Q8).232(C2×C4), (C22×C4).423(C2×C4), (C2×C4).479(C22×C4), (C22×C4○D4).31C2, (C22×C8)2(C2×M4(2)), (C2×C8)2(C22×M4(2)), (C2×C4○D4).342C22, (C22×C8)(C22×M4(2)), (C2×C4)(C2×C8○D4), (C2×C8)(C2×C8○D4), (C2×C8)2(C2×C4○D4), (C22×C8)(C2×C4○D4), (C2×C8)(C22×C4○D4), (C22×C8)(C22×C4○D4), SmallGroup(128,2303)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22×C8○D4
C1C2C4C2×C4C22×C4C23×C4C22×C4○D4 — C22×C8○D4
C1C2 — C22×C8○D4
C1C22×C8 — C22×C8○D4
C1C2C2C4 — C22×C8○D4

Generators and relations for C22×C8○D4
 G = < a,b,c,d,e | a2=b2=c8=e2=1, d2=c4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >

Subgroups: 812 in 752 conjugacy classes, 692 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×12], C4, C4 [×15], C22 [×19], C22 [×36], C8 [×16], C2×C4 [×72], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C2×C8 [×72], M4(2) [×48], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C22×C8, C22×C8 [×39], C2×M4(2) [×36], C8○D4 [×64], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C23×C8 [×3], C22×M4(2) [×3], C2×C8○D4 [×24], C22×C4○D4, C22×C8○D4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C8○D4 [×4], C23×C4 [×30], C25, C2×C8○D4 [×6], C24×C4, C22×C8○D4

Smallest permutation representation of C22×C8○D4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(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 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 51 13 55)(10 52 14 56)(11 53 15 49)(12 54 16 50)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 41 29 45)(26 42 30 46)(27 43 31 47)(28 44 32 48)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)

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

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

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

80 conjugacy classes

class 1 2A···2G2H···2S4A···4H4I···4T8A···8P8Q···8AN
order12···22···24···44···48···88···8
size11···12···21···12···21···12···2

80 irreducible representations

dim111111112
type+++++
imageC1C2C2C2C2C4C4C4C8○D4
kernelC22×C8○D4C23×C8C22×M4(2)C2×C8○D4C22×C4○D4C22×D4C22×Q8C2×C4○D4C22
# reps133241622416

Matrix representation of C22×C8○D4 in GL4(𝔽17) generated by

1000
01600
0010
0001
,
16000
0100
0010
0001
,
4000
01300
0020
0002
,
1000
01600
00016
0010
,
1000
0100
00016
00160
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,13,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,16,0] >;

C22×C8○D4 in GAP, Magma, Sage, TeX

C_2^2\times C_8\circ D_4
% in TeX

G:=Group("C2^2xC8oD4");
// GroupNames label

G:=SmallGroup(128,2303);
// by ID

G=gap.SmallGroup(128,2303);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,723,124]);
// Polycyclic

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

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