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## G = (C2×C12).Q8order 192 = 26·3

### 8th non-split extension by C2×C12 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C12).Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C2×C4.Dic3 — (C2×C12).Q8
 Lower central C3 — C6 — C2×C6 — (C2×C12).Q8
 Upper central C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for (C2×C12).Q8
G = < a,b,c,d | a2=b12=c4=1, d2=ab9c2, ab=ba, cac-1=dad-1=ab6, cbc-1=b7, dbd-1=b5, dcd-1=ab6c-1 >

Subgroups: 184 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, M4(2)⋊4C4, C2×C4.Dic3, C3×C42⋊C2, (C2×C12).Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, M4(2)⋊4C4, C6.C42, (C2×C12).Q8

Smallest permutation representation of (C2×C12).Q8
On 48 points
Generators in S48
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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)
(2 8)(4 10)(6 12)(14 20)(16 22)(18 24)(25 45 31 39)(26 40 32 46)(27 47 33 41)(28 42 34 48)(29 37 35 43)(30 44 36 38)
(1 46 23 35 7 40 17 29)(2 39 24 28 8 45 18 34)(3 44 13 33 9 38 19 27)(4 37 14 26 10 43 20 32)(5 42 15 31 11 48 21 25)(6 47 16 36 12 41 22 30)

G:=sub<Sym(48)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (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), (2,8)(4,10)(6,12)(14,20)(16,22)(18,24)(25,45,31,39)(26,40,32,46)(27,47,33,41)(28,42,34,48)(29,37,35,43)(30,44,36,38), (1,46,23,35,7,40,17,29)(2,39,24,28,8,45,18,34)(3,44,13,33,9,38,19,27)(4,37,14,26,10,43,20,32)(5,42,15,31,11,48,21,25)(6,47,16,36,12,41,22,30)>;

G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (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), (2,8)(4,10)(6,12)(14,20)(16,22)(18,24)(25,45,31,39)(26,40,32,46)(27,47,33,41)(28,42,34,48)(29,37,35,43)(30,44,36,38), (1,46,23,35,7,40,17,29)(2,39,24,28,8,45,18,34)(3,44,13,33,9,38,19,27)(4,37,14,26,10,43,20,32)(5,42,15,31,11,48,21,25)(6,47,16,36,12,41,22,30) );

G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38)], [(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)], [(2,8),(4,10),(6,12),(14,20),(16,22),(18,24),(25,45,31,39),(26,40,32,46),(27,47,33,41),(28,42,34,48),(29,37,35,43),(30,44,36,38)], [(1,46,23,35,7,40,17,29),(2,39,24,28,8,45,18,34),(3,44,13,33,9,38,19,27),(4,37,14,26,10,43,20,32),(5,42,15,31,11,48,21,25),(6,47,16,36,12,41,22,30)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A ··· 8H 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 2 2 2 2 1 1 2 2 2 4 4 4 4 2 2 2 4 4 12 ··· 12 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + - + image C1 C2 C2 C4 C4 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 M4(2)⋊4C4 (C2×C12).Q8 kernel (C2×C12).Q8 C2×C4.Dic3 C3×C42⋊C2 C2×C3⋊C8 C4.Dic3 C3×C22⋊C4 C42⋊C2 C2×C12 C2×C12 C22⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C3 C1 # reps 1 2 1 4 4 4 1 3 1 2 1 2 4 2 4 2 4

Matrix representation of (C2×C12).Q8 in GL4(𝔽73) generated by

 1 0 0 0 7 72 0 0 0 0 72 0 0 0 66 1
,
 49 0 0 0 51 24 0 0 0 0 70 0 0 0 52 3
,
 1 31 0 0 0 72 0 0 0 0 1 31 0 0 7 72
,
 0 0 1 0 0 0 0 1 27 0 0 0 0 27 0 0
G:=sub<GL(4,GF(73))| [1,7,0,0,0,72,0,0,0,0,72,66,0,0,0,1],[49,51,0,0,0,24,0,0,0,0,70,52,0,0,0,3],[1,0,0,0,31,72,0,0,0,0,1,7,0,0,31,72],[0,0,27,0,0,0,0,27,1,0,0,0,0,1,0,0] >;

(C2×C12).Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{12}).Q_8
% in TeX

G:=Group("(C2xC12).Q8");
// GroupNames label

G:=SmallGroup(192,92);
// by ID

G=gap.SmallGroup(192,92);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,184,1123,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a*b^9*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^6,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=a*b^6*c^-1>;
// generators/relations

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