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G = C232M4(2)  order 128 = 27

1st semidirect product of C23 and M4(2) acting via M4(2)/C4=C22

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

Aliases: C232M4(2), (C2×C8)⋊22D4, (C2×C4)⋊3M4(2), C24.57(C2×C4), C4.122C22≀C2, C2.17(C89D4), C2.13(C86D4), (C22×D4).30C4, C22.149(C4×D4), (C22×C4).284D4, C4.186(C4⋊D4), C22.52(C8○D4), (C22×M4(2))⋊9C2, (C22×C8).30C22, C23.78(C22⋊C4), (C2×C42).274C22, C23.313(C22×C4), (C23×C4).252C22, C22.66(C2×M4(2)), C2.18(C24.4C4), (C22×C4).1630C23, C22.7C4240C2, C2.9(C23.23D4), C4.135(C22.D4), (C2×C4×D4).20C2, (C2×C4⋊C4).55C4, (C2×C22⋊C8)⋊37C2, (C2×C4).1529(C2×D4), (C2×C22⋊C4).39C4, (C2×C4).936(C4○D4), (C22×C4).119(C2×C4), (C2×C4).129(C22⋊C4), C22.261(C2×C22⋊C4), C2.27((C22×C8)⋊C2), SmallGroup(128,602)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C232M4(2)
C1C2C4C2×C4C22×C4C23×C4C22×M4(2) — C232M4(2)
C1C23 — C232M4(2)
C1C22×C4 — C232M4(2)
C1C2C2C22×C4 — C232M4(2)

Generators and relations for C232M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 380 in 204 conjugacy classes, 68 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×22], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×10], M4(2) [×8], C22×C4 [×5], C22×C4 [×4], C22×C4 [×10], C2×D4 [×6], C24 [×2], C22⋊C8 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×4], C2×M4(2) [×6], C23×C4 [×2], C22×D4, C22.7C42 [×2], C2×C22⋊C8, C2×C22⋊C8 [×2], C2×C4×D4, C22×M4(2), C232M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×M4(2) [×2], C8○D4 [×2], C23.23D4, C24.4C4, (C22×C8)⋊C2, C89D4 [×2], C86D4 [×2], C232M4(2)

Smallest permutation representation of C232M4(2)
On 64 points
Generators in S64
(1 21)(2 28)(3 23)(4 30)(5 17)(6 32)(7 19)(8 26)(9 33)(10 48)(11 35)(12 42)(13 37)(14 44)(15 39)(16 46)(18 49)(20 51)(22 53)(24 55)(25 50)(27 52)(29 54)(31 56)(34 58)(36 60)(38 62)(40 64)(41 59)(43 61)(45 63)(47 57)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
(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)
(2 6)(4 8)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)(25 57)(26 62)(27 59)(28 64)(29 61)(30 58)(31 63)(32 60)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···144441···14···44···4

44 irreducible representations

dim11111111222222
type+++++++
imageC1C2C2C2C2C4C4C4D4D4M4(2)C4○D4M4(2)C8○D4
kernelC232M4(2)C22.7C42C2×C22⋊C8C2×C4×D4C22×M4(2)C2×C22⋊C4C2×C4⋊C4C22×D4C2×C8C22×C4C2×C4C2×C4C23C22
# reps12311422444448

Matrix representation of C232M4(2) in GL6(𝔽17)

140000
0160000
0016000
0001600
000012
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
1660000
910000
0001600
0013000
0000160
000011
,
140000
0160000
0016000
000100
000010
00001616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,4,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,9,0,0,0,0,6,1,0,0,0,0,0,0,0,13,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C232M4(2) in GAP, Magma, Sage, TeX

C_2^3\rtimes_2M_4(2)
% in TeX

G:=Group("C2^3:2M4(2)");
// GroupNames label

G:=SmallGroup(128,602);
// by ID

G=gap.SmallGroup(128,602);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,e*a*e=a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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