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G = C4⋊Q8⋊15C4  order 128 = 27

10th semidirect product of C4⋊Q8 and C4 acting via C4/C2=C2

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

Aliases: C4⋊Q8⋊15C4, C4.87(C4×D4), C4⋊C4.317D4, (C2×Q8).70D4, C4.6(C4⋊D4), (C22×C4).64D4, C42⋊6C4.6C2, C42.148(C2×C4), C23.561(C2×D4), C2.5(D4.10D4), C22.100C22≀C2, (C2×C42).285C22, (C22×C4).684C23, C23.38D4.4C2, (C22×Q8).16C22, C42⋊C2.21C22, C2.25(C23.23D4), (C2×M4(2)).181C22, C23.32C23.3C2, C22.49(C22.D4), (C2×C4⋊Q8).8C2, (C2×Q8).67(C2×C4), (C2×C4).56(C4○D4), (C2×C4).1005(C2×D4), (C2×C4).13(C22⋊C4), (C2×C4).186(C22×C4), (C2×C4.10D4).8C2, C22.41(C2×C22⋊C4), SmallGroup(128,618)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C4⋊Q8⋊15C4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.32C23 — C4⋊Q8⋊15C4
Lower central
C1 — C2 — C2×C4 — C4⋊Q8⋊15C4
Upper central
C1 — C22 — C22×C4 — C4⋊Q8⋊15C4
Jennings
C1 — C2 — C2 — C22×C4 — C4⋊Q8⋊15C4

Generators and relations for C4⋊Q8⋊15C4
 G = < a,b,c,d | a4=b4=d4=1, c2=b2, dad-1=ab=ba, cac-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c >

Subgroups: 292 in 162 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C4.10D4, Q8⋊C4, C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×Q8, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C42⋊6C4, C2×C4.10D4, C23.38D4, C23.32C23, C2×C4⋊Q8, C4⋊Q8⋊15C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, D4.10D4, C4⋊Q8⋊15C4

Smallest permutation representation of C4⋊Q8⋊15C4
►On 32 points
Generators in S32
(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 15 9 7)(2 16 10 8)(3 12 14 5)(4 11 13 6)(17 24 19 22)(18 21 20 23)(25 31 27 29)(26 32 28 30)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4T4U4V8A8B8C8D
order12222244444···4448888
size11112222224···4888888

32 irreducible representations

dim111111122224
type+++++++++-
imageC1C2C2C2C2C2C4D4D4D4C4○D4D4.10D4
kernelC4⋊Q8⋊15C4C42⋊6C4C2×C4.10D4C23.38D4C23.32C23C2×C4⋊Q8C4⋊Q8C4⋊C4C22×C4C2×Q8C2×C4C2
# reps121211842244

Matrix representation of C4⋊Q8⋊15C4 ►in GL6(𝔽17)

0160000
100000
0016000
0001600
000001
0000160
,
100000
010000
0001600
001000
000001
0000160
,
0160000
1600000
001700
0071600
00001610
0000101
,
1300000
0130000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,7,0,0,0,0,7,16,0,0,0,0,0,0,16,10,0,0,0,0,10,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C4⋊Q8⋊15C4 in GAP, Magma, Sage, TeX 

C_4\rtimes Q_8\rtimes_{15}C_4
 
% in TeX
 
G:=Group("C4:Q8:15C4");
 
// GroupNames label
 
G:=SmallGroup(128,618);
 
// by ID
 
G=gap.SmallGroup(128,618);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,521,248,2804,1411,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^4=1,c^2=b^2,d*a*d^-1=a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c>;
 
// generators/relations
 

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