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## G = (C2×Q8)⋊C12order 192 = 26·3

### The semidirect product of C2×Q8 and C12 acting via C12/C2=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — (C2×Q8)⋊C12
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C22×SL2(𝔽3) — (C2×Q8)⋊C12
 Lower central Q8 — (C2×Q8)⋊C12
 Upper central C1 — C22 — C22⋊C4

Generators and relations for (C2×Q8)⋊C12
G = < a,b,c,d | a2=b4=d12=1, c2=b2, ab=ba, ac=ca, dad-1=ab2, cbc-1=b-1, dbd-1=b2c, dcd-1=abc >

Subgroups: 245 in 86 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C2×C4, C2×C4, Q8, Q8, C23, C12, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, SL2(𝔽3), C2×C12, C22×C6, C42⋊C2, C4×Q8, C4×Q8, C22×Q8, C3×C22⋊C4, C2×SL2(𝔽3), C2×SL2(𝔽3), C23.32C23, C4×SL2(𝔽3), C22×SL2(𝔽3), (C2×Q8)⋊C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C22×A4, C2×C4×A4, D4.A4, (C2×Q8)⋊C12

Smallest permutation representation of (C2×Q8)⋊C12
On 32 points
Generators in S32
(1 7)(2 4)(3 5)(6 8)(9 15)(10 30)(11 17)(12 32)(13 19)(14 22)(16 24)(18 26)(20 28)(21 27)(23 29)(25 31)
(1 12 5 26)(2 23 6 9)(3 18 7 32)(4 29 8 15)(10 20 24 22)(11 13 25 27)(14 30 28 16)(17 19 31 21)
(1 20 5 22)(2 31 6 17)(3 14 7 28)(4 25 8 11)(9 21 23 19)(10 26 24 12)(13 15 27 29)(16 32 30 18)
(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)

G:=sub<Sym(32)| (1,7)(2,4)(3,5)(6,8)(9,15)(10,30)(11,17)(12,32)(13,19)(14,22)(16,24)(18,26)(20,28)(21,27)(23,29)(25,31), (1,12,5,26)(2,23,6,9)(3,18,7,32)(4,29,8,15)(10,20,24,22)(11,13,25,27)(14,30,28,16)(17,19,31,21), (1,20,5,22)(2,31,6,17)(3,14,7,28)(4,25,8,11)(9,21,23,19)(10,26,24,12)(13,15,27,29)(16,32,30,18), (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)>;

G:=Group( (1,7)(2,4)(3,5)(6,8)(9,15)(10,30)(11,17)(12,32)(13,19)(14,22)(16,24)(18,26)(20,28)(21,27)(23,29)(25,31), (1,12,5,26)(2,23,6,9)(3,18,7,32)(4,29,8,15)(10,20,24,22)(11,13,25,27)(14,30,28,16)(17,19,31,21), (1,20,5,22)(2,31,6,17)(3,14,7,28)(4,25,8,11)(9,21,23,19)(10,26,24,12)(13,15,27,29)(16,32,30,18), (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) );

G=PermutationGroup([[(1,7),(2,4),(3,5),(6,8),(9,15),(10,30),(11,17),(12,32),(13,19),(14,22),(16,24),(18,26),(20,28),(21,27),(23,29),(25,31)], [(1,12,5,26),(2,23,6,9),(3,18,7,32),(4,29,8,15),(10,20,24,22),(11,13,25,27),(14,30,28,16),(17,19,31,21)], [(1,20,5,22),(2,31,6,17),(3,14,7,28),(4,25,8,11),(9,21,23,19),(10,26,24,12),(13,15,27,29),(16,32,30,18)], [(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)]])

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 4 4 type + + + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 A4 C2×A4 C2×A4 C4×A4 D4.A4 D4.A4 kernel (C2×Q8)⋊C12 C4×SL2(𝔽3) C22×SL2(𝔽3) C23.32C23 C2×SL2(𝔽3) C4×Q8 C22×Q8 C2×Q8 C22⋊C4 C2×C4 C23 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 4 2 4

Matrix representation of (C2×Q8)⋊C12 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 5 8 0 1
,
 12 0 0 0 0 0 0 12 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 8 5 1 11 0 0 0 0 5 1 12
,
 0 12 1 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 9 10 0 0 0 0 0 10 4 0 0 0 0 0 11 2 7 7 0 0 0 11 9 4 6
,
 0 8 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 11 2 7 7 0 0 0 1 0 0 0 0 0 0 0 10 2 11

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,5,0,0,0,0,12,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,12,0,5,5,0,0,0,0,0,1,1,0,0,0,0,0,11,12],[0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,10,11,11,0,0,0,10,4,2,9,0,0,0,0,0,7,4,0,0,0,0,0,7,6],[0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,2,0,10,0,0,0,1,7,0,2,0,0,0,0,7,0,11] >;

(C2×Q8)⋊C12 in GAP, Magma, Sage, TeX

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

G:=Group("(C2xQ8):C12");
// GroupNames label

G:=SmallGroup(192,998);
// by ID

G=gap.SmallGroup(192,998);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,1373,92,438,172,775,285,124]);
// Polycyclic

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

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