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G = C42.42Q8order 128 = 27

2nd non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Aliases: C42.42Q8, C42.456D4, C42.621C23, (C2xC8):6C8, C8.17(C2xC8), (C4xC8).26C4, C4o2(C8:2C8), C4o2(C8:1C8), C8:1C8:31C2, C8:2C8:31C2, C4.27(C4:C8), C42o(C8:1C8), C42o(C8:2C8), C22.5(C4:C8), (C22xC8).36C4, C4.26(C22xC8), C4.122(C4oD8), (C22xC4).74Q8, C4:C8.265C22, C23.47(C4:C4), C42.309(C2xC4), (C4xC8).389C22, (C22xC4).541D4, C4.41(C2xM4(2)), (C2xC4).75M4(2), C4.14(C8.C4), C42.12C4.28C2, (C2xC42).1040C22, C2.1(C23.25D4), C2.6(C2xC4:C8), (C2xC4xC8).41C2, (C2xC4).82(C2xC8), (C2xC8).219(C2xC4), C2.3(C2xC8.C4), C22.47(C2xC4:C4), (C2xC4).148(C2xQ8), (C2xC4).161(C4:C4), (C2xC4).1457(C2xD4), (C22xC4).473(C2xC4), (C2xC4).503(C22xC4), SmallGroup(128,296)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.42Q8
C1C2C22C2xC4C42C2xC42C2xC4xC8 — C42.42Q8
C1C2C4 — C42.42Q8
C1C42C2xC42 — C42.42Q8
C1C22C22C42 — C42.42Q8

Generators and relations for C42.42Q8
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 124 in 90 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, C2xC8, C22xC4, C4xC8, C4xC8, C22:C8, C4:C8, C2xC42, C22xC8, C8:2C8, C8:1C8, C2xC4xC8, C42.12C4, C42.42Q8
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, Q8, C23, C4:C4, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4:C8, C8.C4, C2xC4:C4, C22xC8, C2xM4(2), C4oD8, C2xC4:C8, C23.25D4, C2xC8.C4, C42.42Q8

Smallest permutation representation of C42.42Q8
On 64 points
Generators in S64
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)
(1 33 41 55)(2 34 42 56)(3 35 43 49)(4 36 44 50)(5 37 45 51)(6 38 46 52)(7 39 47 53)(8 40 48 54)(9 57 25 23)(10 58 26 24)(11 59 27 17)(12 60 28 18)(13 61 29 19)(14 62 30 20)(15 63 31 21)(16 64 32 22)
(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 32 39 20 45 12 49 58)(2 27 40 23 46 15 50 61)(3 30 33 18 47 10 51 64)(4 25 34 21 48 13 52 59)(5 28 35 24 41 16 53 62)(6 31 36 19 42 11 54 57)(7 26 37 22 43 14 55 60)(8 29 38 17 44 9 56 63)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8AF
order1222224···44···48···88···8
size1111221···12···22···24···4

56 irreducible representations

dim111111112222222
type++++++-+-
imageC1C2C2C2C2C4C4C8D4Q8D4Q8M4(2)C8.C4C4oD8
kernelC42.42Q8C8:2C8C8:1C8C2xC4xC8C42.12C4C4xC8C22xC8C2xC8C42C42C22xC4C22xC4C2xC4C4C4
# reps1221244161111488

Matrix representation of C42.42Q8 in GL3(F17) generated by

1600
0130
0013
,
400
040
0013
,
100
0150
009
,
1500
001
010
G:=sub<GL(3,GF(17))| [16,0,0,0,13,0,0,0,13],[4,0,0,0,4,0,0,0,13],[1,0,0,0,15,0,0,0,9],[15,0,0,0,0,1,0,1,0] >;

C42.42Q8 in GAP, Magma, Sage, TeX

C_4^2._{42}Q_8
% in TeX

G:=Group("C4^2.42Q8");
// GroupNames label

G:=SmallGroup(128,296);
// by ID

G=gap.SmallGroup(128,296);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,184,1123,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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