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G = C8:M4(2)  order 128 = 27

2nd semidirect product of C8 and M4(2) acting via M4(2)/C4=C22

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

Aliases: C8:2M4(2), C42.649C23, C8:C8:7C2, Q8:C8:42C2, C8:1C8:18C2, C4.Q8.6C4, D4:C8.16C2, C8:4Q8:31C2, (C2xC8).183D4, C4.16(C8oD4), C8:6D4.14C2, C2.9(C8:6D4), (C4xSD16).4C2, (C2xSD16).2C4, D4:C4.11C4, C4:C8.227C22, (C4xC8).318C22, Q8:C4.11C4, (C4xD4).15C22, C22.140(C4xD4), C4.10(C2xM4(2)), (C4xQ8).15C22, C2.10(C8.26D4), C4.146(C8:C22), C2.7(SD16:C4), C4.140(C8.C22), C4:C4.62(C2xC4), (C2xC8).57(C2xC4), (C2xD4).60(C2xC4), (C2xQ8).55(C2xC4), (C2xC4).1485(C2xD4), (C2xC4).510(C4oD4), (C2xC4).341(C22xC4), SmallGroup(128,324)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C8:M4(2)
C1C2C22C2xC4C42C4xC8C8:6D4 — C8:M4(2)
C1C2C2xC4 — C8:M4(2)
C1C2xC4C4xC8 — C8:M4(2)
C1C22C22C42 — C8:M4(2)

Generators and relations for C8:M4(2)
 G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a3, cbc=b5 >

Subgroups: 152 in 81 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), SD16, C22xC4, C2xD4, C2xQ8, C4xC8, C8:C4, C22:C8, D4:C4, Q8:C4, C4:C8, C4:C8, C4.Q8, C4xD4, C4xQ8, C2xM4(2), C2xSD16, C8:C8, D4:C8, Q8:C8, C8:1C8, C8:6D4, C4xSD16, C8:4Q8, C8:M4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), C22xC4, C2xD4, C4oD4, C4xD4, C2xM4(2), C8oD4, C8:C22, C8.C22, C8:6D4, SD16:C4, C8.26D4, C8:M4(2)

Smallest permutation representation of C8:M4(2)
On 64 points
Generators in S64
(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 31 61 15 38 22 56 45)(2 30 62 14 39 21 49 44)(3 29 63 13 40 20 50 43)(4 28 64 12 33 19 51 42)(5 27 57 11 34 18 52 41)(6 26 58 10 35 17 53 48)(7 25 59 9 36 24 54 47)(8 32 60 16 37 23 55 46)
(2 4)(3 7)(6 8)(9 43)(10 46)(11 41)(12 44)(13 47)(14 42)(15 45)(16 48)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(33 39)(35 37)(36 40)(49 51)(50 54)(53 55)(58 60)(59 63)(62 64)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K8A···8L8M8N8O8P
order12222444444444448···88888
size11118111122228884···48888

32 irreducible representations

dim1111111111112222444
type++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4M4(2)C4oD4C8oD4C8:C22C8.C22C8.26D4
kernelC8:M4(2)C8:C8D4:C8Q8:C8C8:1C8C8:6D4C4xSD16C8:4Q8D4:C4Q8:C4C4.Q8C2xSD16C2xC8C8C2xC4C4C4C4C2
# reps1111111122222424112

Matrix representation of C8:M4(2) in GL6(F17)

100000
010000
0006015
00141112
0002011
00161536
,
010000
1300000
000010
00001616
001000
00161600
,
1600000
010000
001000
00161600
000010
00001616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,16,0,0,6,11,2,15,0,0,0,1,0,3,0,0,15,2,11,6],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,1,16,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C8:M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes M_4(2)
% in TeX

G:=Group("C8:M4(2)");
// GroupNames label

G:=SmallGroup(128,324);
// by ID

G=gap.SmallGroup(128,324);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,723,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^5>;
// generators/relations

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