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## G = (C2×D4).24Q8order 128 = 27

### 5th non-split extension by C2×D4 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 — (C2×D4).24Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×D4).24Q8
 Lower central C1 — C2 — C2×C4 — (C2×D4).24Q8
 Upper central C1 — C4 — C2×M4(2) — (C2×D4).24Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×D4).24Q8

Generators and relations for (C2×D4).24Q8
G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=abd2, ab=ba, ece-1=ac=ca, ad=da, eae-1=ab2, cbc=b-1, bd=db, be=eb, cd=dc, ede-1=d3 >

Subgroups: 244 in 130 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C4.Q8, C2.D8, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C4.9C42, C23.C23, C23.25D4, C2×C8○D4, (C2×D4).24Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, (C2×D4).24Q8

Smallest permutation representation of (C2×D4).24Q8
On 32 points
Generators in S32
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(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)
(1 11 31 17)(2 14 32 20)(3 9 25 23)(4 12 26 18)(5 15 27 21)(6 10 28 24)(7 13 29 19)(8 16 30 22)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 8 ··· 8 2 2 2 2 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - - image C1 C2 C2 C2 C2 C4 C4 D4 D4 Q8 Q8 C4○D4 (C2×D4).24Q8 kernel (C2×D4).24Q8 C4.9C42 C23.C23 C23.25D4 C2×C8○D4 C22×C8 C2×M4(2) C2×C8 C2×D4 C2×D4 C2×Q8 C2×C4 C1 # reps 1 2 2 2 1 4 4 4 2 1 1 4 4

Matrix representation of (C2×D4).24Q8 in GL4(𝔽17) generated by

 16 2 0 0 0 1 0 0 0 13 0 16 0 13 16 0
,
 4 9 0 0 0 13 0 0 16 1 0 13 16 1 13 0
,
 4 9 15 0 0 0 16 1 16 1 0 13 16 2 0 13
,
 15 11 0 0 0 9 0 0 0 12 12 3 0 12 3 12
,
 1 15 8 0 1 15 4 4 13 8 0 1 0 4 0 1
G:=sub<GL(4,GF(17))| [16,0,0,0,2,1,13,13,0,0,0,16,0,0,16,0],[4,0,16,16,9,13,1,1,0,0,0,13,0,0,13,0],[4,0,16,16,9,0,1,2,15,16,0,0,0,1,13,13],[15,0,0,0,11,9,12,12,0,0,12,3,0,0,3,12],[1,1,13,0,15,15,8,4,8,4,0,0,0,4,1,1] >;

(C2×D4).24Q8 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{24}Q_8
% in TeX

G:=Group("(C2xD4).24Q8");
// GroupNames label

G:=SmallGroup(128,544);
// by ID

G=gap.SmallGroup(128,544);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,2804,1027]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=b^2,e^2=a*b*d^2,a*b=b*a,e*c*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*d*e^-1=d^3>;
// generators/relations

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