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## G = C42.43Q8order 128 = 27

### 3rd non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.43Q8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C42.43Q8
 Lower central C1 — C2 — C2×C4 — C42.43Q8
 Upper central C1 — C2×C4 — C2×C42 — C42.43Q8
 Jennings C1 — C22 — C22 — C42 — C42.43Q8

Generators and relations for C42.43Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 124 in 84 conjugacy classes, 52 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], C22×C4 [×3], C4×C8 [×4], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C82C8 [×2], C81C8 [×2], C2×C4×C8, C42.6C4 [×2], C42.43Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, C8.C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C4○D8 [×2], C4⋊M4(2), C23.25D4, C2×C8.C4, C42.43Q8

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D4 Q8 M4(2) C4○D8 C8.C4 kernel C42.43Q8 C8⋊2C8 C8⋊1C8 C2×C4×C8 C42.6C4 C4×C8 C22×C8 C42 C42 C22×C4 C22×C4 C8 C4 C22 # reps 1 2 2 1 2 4 4 1 1 1 1 8 8 8

Matrix representation of C42.43Q8 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 16 0 0 0 0 1
,
 13 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 8 0 0 0 0 2 0 0 0 0 4 0 0 0 0 13
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 13 0
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[8,0,0,0,0,2,0,0,0,0,4,0,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,0,13,0,0,1,0] >;

C42.43Q8 in GAP, Magma, Sage, TeX

C_4^2._{43}Q_8
% in TeX

G:=Group("C4^2.43Q8");
// GroupNames label

G:=SmallGroup(128,300);
// by ID

G=gap.SmallGroup(128,300);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1430,184,1123,136,172]);
// 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,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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