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G = C42.Q8order 128 = 27

19th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.19Q8, C42.377D4, C42.629C23, C4⋊C8.14C4, C81C828C2, C82C827C2, C4.45(C8○D4), C22⋊C8.12C4, C4.124(C4○D8), (C22×C4).12Q8, C4⋊C8.269C22, C23.20(C4⋊C4), C42.125(C2×C4), (C4×C8).237C22, (C22×C4).250D4, C42.6C4.30C2, (C2×C42).228C22, C2.7(M4(2).C4), C42.12C4.30C2, C2.6(C23.25D4), C2.7(C42.6C22), (C2×C8).28(C2×C4), (C2×C4).36(C4⋊C4), C22.86(C2×C4⋊C4), (C2×C4).156(C2×Q8), (C2×C4).1465(C2×D4), (C2×C4).511(C22×C4), (C22×C4).250(C2×C4), SmallGroup(128,304)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.Q8
C1C2C22C2×C4C42C2×C42C42.12C4 — C42.Q8
C1C2C2×C4 — C42.Q8
C1C2×C4C2×C42 — C42.Q8
C1C22C22C42 — C42.Q8

Generators and relations for C42.Q8
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, cac-1=a-1b2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 116 in 75 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×8], C2×C4 [×6], C2×C4 [×5], C23, C42 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C4×C8 [×2], C4×C8, C8⋊C4, C22⋊C8 [×2], C22⋊C8 [×2], C4⋊C8 [×6], C2×C42, C82C8 [×2], C81C8 [×2], C42.12C4 [×2], C42.6C4, C42.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C8○D4 [×2], C4○D8 [×2], C42.6C22, C23.25D4, M4(2).C4, C42.Q8

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4L4M8A···8P8Q8R8S8T
order1222244444···448···88888
size1111411112···244···48888

38 irreducible representations

dim11111112222224
type++++++-+-
imageC1C2C2C2C2C4C4D4Q8D4Q8C8○D4C4○D8M4(2).C4
kernelC42.Q8C82C8C81C8C42.12C4C42.6C4C22⋊C8C4⋊C8C42C42C22×C4C22×C4C4C4C2
# reps12221441111882

Matrix representation of C42.Q8 in GL4(𝔽17) generated by

4000
0400
00160
00161
,
4000
111300
0040
0004
,
8000
4200
00115
00116
,
71500
81000
0090
0098
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,16,0,0,0,1],[4,11,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[8,4,0,0,0,2,0,0,0,0,1,1,0,0,15,16],[7,8,0,0,15,10,0,0,0,0,9,9,0,0,0,8] >;

C42.Q8 in GAP, Magma, Sage, TeX

C_4^2.Q_8
% in TeX

G:=Group("C4^2.Q8");
// GroupNames label

G:=SmallGroup(128,304);
// by ID

G=gap.SmallGroup(128,304);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,723,184,1123,136,172]);
// 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,c*a*c^-1=a^-1*b^2,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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