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G = C22xC8oD4order 128 = 27

Direct product of C22 and C8oD4

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

Aliases: C22xC8oD4, C8.21C24, C4.20C25, M4(2):13C23, D4o(C22xC8), C8o(C22xD4), Q8o(C22xC8), C8o(C22xQ8), (C23xC8):17C2, (C2xC8):18C23, C4.66(C23xC4), C2.14(C24xC4), C24.102(C2xC4), (C2xC4).604C24, (C22xC8):72C22, C4oD4.35C23, (C22xD4).47C4, D4.26(C22xC4), M4(2)o2(C22xC8), C8o2(C22xM4(2)), C22.7(C23xC4), (C22xQ8).37C4, Q8.27(C22xC4), (C2xM4(2)):82C22, (C22xM4(2)):29C2, (C23xC4).710C22, C23.156(C22xC4), (C22xC4).1588C23, C4o(C2xC8oD4), C8o2(C2xC4oD4), C8o2(C2xC8oD4), (C2xC8)o2(C2xD4), (C2xC8)o2(C2xQ8), (C2xC4)o(C8oD4), C8o(C22xC4oD4), C4oD4o(C22xC8), (C2xC8)o2(C8oD4), (C2xC8)o2(C4oD4), (C2xC8)o(C22xQ8), (C2xQ8)o(C22xC8), C4oD4.40(C2xC4), (C2xC4oD4).36C4, (C2xC8)o3(C2xM4(2)), (C2xD4).254(C2xC4), (C22xC8)o(C22xQ8), (C2xQ8).232(C2xC4), (C22xC4).423(C2xC4), (C2xC4).479(C22xC4), (C22xC4oD4).31C2, (C22xC8)o2(C2xM4(2)), (C2xC8)o2(C22xM4(2)), (C2xC4oD4).342C22, (C22xC8)o(C22xM4(2)), (C2xC4)o(C2xC8oD4), (C2xC8)o(C2xC8oD4), (C2xC8)o2(C2xC4oD4), (C22xC8)o(C2xC4oD4), (C2xC8)o(C22xC4oD4), (C22xC8)o(C22xC4oD4), SmallGroup(128,2303)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22xC8oD4
C1C2C4C2xC4C22xC4C23xC4C22xC4oD4 — C22xC8oD4
C1C2 — C22xC8oD4
C1C22xC8 — C22xC8oD4
C1C2C2C4 — C22xC8oD4

Generators and relations for C22xC8oD4
 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, C2, C4, C4, C22, C22, C8, C2xC4, D4, Q8, C23, C23, C23, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C22xC8, C22xC8, C2xM4(2), C8oD4, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C23xC8, C22xM4(2), C2xC8oD4, C22xC4oD4, C22xC8oD4
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C24, C8oD4, C23xC4, C25, C2xC8oD4, C24xC4, C22xC8oD4

Smallest permutation representation of C22xC8oD4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(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 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 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 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)(25 47 29 43)(26 48 30 44)(27 41 31 45)(28 42 32 46)(33 51 37 55)(34 52 38 56)(35 53 39 49)(36 54 40 50)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(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)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46)(33,51,37,55)(34,52,38,56)(35,53,39,49)(36,54,40,50), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(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)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46)(33,51,37,55)(34,52,38,56)(35,53,39,49)(36,54,40,50), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(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),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(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)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,57),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(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)], [(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17),(25,47,29,43),(26,48,30,44),(27,41,31,45),(28,42,32,46),(33,51,37,55),(34,52,38,56),(35,53,39,49),(36,54,40,50)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(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+++++
imageC1C2C2C2C2C4C4C4C8oD4
kernelC22xC8oD4C23xC8C22xM4(2)C2xC8oD4C22xC4oD4C22xD4C22xQ8C2xC4oD4C22
# reps133241622416

Matrix representation of C22xC8oD4 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] >;

C22xC8oD4 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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