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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, (C2×C8)⋊6C8, C8.17(C2×C8), (C4×C8).26C4, C42(C82C8), C42(C81C8), C81C831C2, C82C831C2, C4.27(C4⋊C8), C42(C81C8), C42(C82C8), C22.5(C4⋊C8), (C22×C8).36C4, C4.26(C22×C8), C4.122(C4○D8), (C22×C4).74Q8, C4⋊C8.265C22, C23.47(C4⋊C4), C42.309(C2×C4), (C4×C8).389C22, (C22×C4).541D4, C4.41(C2×M4(2)), (C2×C4).75M4(2), C4.14(C8.C4), C42.12C4.28C2, (C2×C42).1040C22, C2.1(C23.25D4), C2.6(C2×C4⋊C8), (C2×C4×C8).41C2, (C2×C4).82(C2×C8), (C2×C8).219(C2×C4), C2.3(C2×C8.C4), C22.47(C2×C4⋊C4), (C2×C4).148(C2×Q8), (C2×C4).161(C4⋊C4), (C2×C4).1457(C2×D4), (C22×C4).473(C2×C4), (C2×C4).503(C22×C4), SmallGroup(128,296)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.42Q8
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C42.42Q8
C1C2C4 — C42.42Q8
C1C42C2×C42 — 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 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×4], C2×C8 [×4], C2×C8 [×6], C22×C4 [×3], C4×C8 [×4], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C82C8 [×2], C81C8 [×2], C2×C4×C8, C42.12C4 [×2], C42.42Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C8.C4 [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C4○D8 [×2], C2×C4⋊C8, C23.25D4, C2×C8.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 46 53)(2 34 47 54)(3 35 48 55)(4 36 41 56)(5 37 42 49)(6 38 43 50)(7 39 44 51)(8 40 45 52)(9 64 21 29)(10 57 22 30)(11 58 23 31)(12 59 24 32)(13 60 17 25)(14 61 18 26)(15 62 19 27)(16 63 20 28)
(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 10 42 63 55 18)(2 27 40 13 43 58 56 21)(3 30 33 16 44 61 49 24)(4 25 34 11 45 64 50 19)(5 28 35 14 46 59 51 22)(6 31 36 9 47 62 52 17)(7 26 37 12 48 57 53 20)(8 29 38 15 41 60 54 23)

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

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

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

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.C4C4○D8
kernelC42.42Q8C82C8C81C8C2×C4×C8C42.12C4C4×C8C22×C8C2×C8C42C42C22×C4C22×C4C2×C4C4C4
# 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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