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G = C22×C42⋊C2order 128 = 27

Direct product of C22 and C42⋊C2

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

Aliases: C22×C42⋊C2, C4217C23, C22.7C25, C25.93C22, C23.102C24, C24.654C23, C4⋊C422C23, (C23×C4)⋊19C4, C2.3(C24×C4), (C24×C4).13C2, C4.65(C23×C4), (C22×C42)⋊9C2, C24.131(C2×C4), (C2×C4).154C24, (C2×C42)⋊83C22, C22.47(C23×C4), C23.374(C4○D4), C22⋊C4.123C23, (C23×C4).704C22, C23.233(C22×C4), (C22×C4).1291C23, C4(C22×C4⋊C4), C4⋊C42(C22×C4), C4(C2×C42⋊C2), C4(C22×C22⋊C4), (C22×C4⋊C4)⋊48C2, (C2×C4)⋊13(C22×C4), (C22×C4)⋊53(C2×C4), C22⋊C42(C22×C4), C2.1(C22×C4○D4), (C2×C4⋊C4)⋊143C22, (C2×C4)2(C42⋊C2), C22.142(C2×C4○D4), (C22×C4)(C42⋊C2), (C22×C22⋊C4).31C2, (C2×C22⋊C4).559C22, (C2×C4)4(C2×C4⋊C4), (C2×C4)(C22×C4⋊C4), (C2×C4)3(C2×C22⋊C4), (C22×C4)2(C2×C4⋊C4), (C2×C4)(C2×C42⋊C2), (C22×C4)(C22×C4⋊C4), (C22×C4)(C2×C42⋊C2), SmallGroup(128,2153)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22×C42⋊C2
C1C2C22C23C24C23×C4C24×C4 — C22×C42⋊C2
C1C2 — C22×C42⋊C2
C1C23×C4 — C22×C42⋊C2
C1C22 — C22×C42⋊C2

Generators and relations for C22×C42⋊C2
 G = < a,b,c,d,e | a2=b2=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, de=ed >

Subgroups: 1148 in 940 conjugacy classes, 732 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×16], C4 [×16], C22, C22 [×42], C22 [×56], C2×C4 [×136], C2×C4 [×48], C23 [×43], C23 [×56], C42 [×32], C22⋊C4 [×32], C4⋊C4 [×32], C22×C4 [×164], C22×C4 [×16], C24, C24 [×14], C24 [×8], C2×C42 [×24], C2×C22⋊C4 [×24], C2×C4⋊C4 [×24], C42⋊C2 [×64], C23×C4 [×2], C23×C4 [×32], C25, C22×C42 [×2], C22×C22⋊C4 [×2], C22×C4⋊C4 [×2], C2×C42⋊C2 [×24], C24×C4, C22×C42⋊C2
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C4○D4 [×8], C24 [×31], C42⋊C2 [×16], C23×C4 [×30], C2×C4○D4 [×12], C25, C2×C42⋊C2 [×12], C24×C4, C22×C4○D4 [×2], C22×C42⋊C2

Smallest permutation representation of C22×C42⋊C2
On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 63)(6 64)(7 61)(8 62)(9 17)(10 18)(11 19)(12 20)(13 60)(14 57)(15 58)(16 59)(21 47)(22 48)(23 45)(24 46)(25 51)(26 52)(27 49)(28 50)(33 56)(34 53)(35 54)(36 55)(37 41)(38 42)(39 43)(40 44)
(1 27)(2 28)(3 25)(4 26)(5 37)(6 38)(7 39)(8 40)(9 34)(10 35)(11 36)(12 33)(13 46)(14 47)(15 48)(16 45)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(29 49)(30 50)(31 51)(32 52)(41 63)(42 64)(43 61)(44 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 59 55 39)(2 60 56 40)(3 57 53 37)(4 58 54 38)(5 25 21 17)(6 26 22 18)(7 27 23 19)(8 28 24 20)(9 63 51 47)(10 64 52 48)(11 61 49 45)(12 62 50 46)(13 33 44 30)(14 34 41 31)(15 35 42 32)(16 36 43 29)
(1 27)(2 20)(3 25)(4 18)(5 37)(6 58)(7 39)(8 60)(9 34)(10 32)(11 36)(12 30)(13 62)(14 47)(15 64)(16 45)(17 53)(19 55)(21 57)(22 38)(23 59)(24 40)(26 54)(28 56)(29 49)(31 51)(33 50)(35 52)(41 63)(42 48)(43 61)(44 46)

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

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

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

80 conjugacy classes

class 1 2A···2O2P···2W4A···4P4Q···4BD
order12···22···24···44···4
size11···12···21···12···2

80 irreducible representations

dim11111112
type++++++
imageC1C2C2C2C2C2C4C4○D4
kernelC22×C42⋊C2C22×C42C22×C22⋊C4C22×C4⋊C4C2×C42⋊C2C24×C4C23×C4C23
# reps12222413216

Matrix representation of C22×C42⋊C2 in GL5(𝔽5)

40000
04000
00400
00010
00001
,
10000
04000
00100
00040
00004
,
20000
04000
00100
00001
00010
,
40000
04000
00100
00020
00002
,
40000
01000
00400
00040
00001

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1] >;

C22×C42⋊C2 in GAP, Magma, Sage, TeX

C_2^2\times C_4^2\rtimes C_2
% in TeX

G:=Group("C2^2xC4^2:C2");
// GroupNames label

G:=SmallGroup(128,2153);
// by ID

G=gap.SmallGroup(128,2153);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,184]);
// Polycyclic

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

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