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G = (C2×C8)⋊11D4  order 128 = 27

7th semidirect product of C2×C8 and D4 acting via D4/C2=C22

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

Aliases: (C2×C8)⋊11D4, C8⋊D4⋊3C2, C8⋊8D4⋊43C2, C8.119(C2×D4), (C2×D4).216D4, C4⋊C4.26C23, (C2×Q8).171D4, C23.75(C2×D4), C2.D8⋊70C22, C4.Q8⋊56C22, C22⋊Q8⋊3C22, C4.69(C4⋊D4), (C2×C8).253C23, (C2×C4).261C24, (C22×C8)⋊41C22, (C22×SD16)⋊2C2, (C2×D4).64C23, C4.155(C22×D4), (C2×Q8).52C23, C2.16(D4○SD16), Q8⋊C4⋊53C22, C4⋊D4.19C22, C23.25D4⋊27C2, C23.38D4⋊34C2, C23.37D4⋊34C2, C22.86(C4⋊D4), (C2×M4(2))⋊53C22, (C22×C4).983C23, C22.521(C22×D4), D4⋊C4.129C22, C22.29C24.11C2, (C2×SD16).134C22, (C22×D4).349C22, (C22×Q8).282C22, C42⋊C2.110C22, C23.38C23⋊10C2, (C2×C8○D4)⋊1C2, C4.28(C2×C4○D4), (C2×C4).129(C2×D4), C2.79(C2×C4⋊D4), (C2×C4).282(C4○D4), (C2×C4○D4).301C22, SmallGroup(128,1789)

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

Derived series
C1 — C2×C4 — (C2×C8)⋊11D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊11D4
Lower central
C1 — C2 — C2×C4 — (C2×C8)⋊11D4
Upper central
C1 — C22 — C2×C4○D4 — (C2×C8)⋊11D4
Jennings
C1 — C2 — C2 — C2×C4 — (C2×C8)⋊11D4

Generators and relations for (C2×C8)⋊11D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=b3, dcd=c-1 >

Subgroups: 508 in 246 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×SD16, C2×SD16, C22×D4, C22×Q8, C2×C4○D4, C23.37D4, C23.38D4, C23.25D4, C8⋊8D4, C8⋊D4, C22.29C24, C23.38C23, C2×C8○D4, C22×SD16, (C2×C8)⋊11D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○SD16, (C2×C8)⋊11D4

Smallest permutation representation of (C2×C8)⋊11D4
►On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 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)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4L8A8B8C8D8E···8J
order12222222224444444···488888···8
size11112244882222448···822224···4

32 irreducible representations

dim111111111122224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○SD16
kernel(C2×C8)⋊11D4C23.37D4C23.38D4C23.25D4C8⋊8D4C8⋊D4C22.29C24C23.38C23C2×C8○D4C22×SD16C2×C8C2×D4C2×Q8C2×C4C2
# reps111144111143144

Matrix representation of (C2×C8)⋊11D4 ►in GL6(𝔽17)

100000
010000
001000
000100
0000160
00116016
,
040000
400000
0012500
00121200
0051207
005057
,
0160000
100000
000010
00161162
001000
0010116
,
100000
0160000
001000
0001600
000010
0010116

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,12,12,5,5,0,0,5,12,12,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,1,0,0,0,1,0,0,0,0,1,16,0,1,0,0,0,2,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
 

(C2×C8)⋊11D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8)\rtimes_{11}D_4
 
% in TeX
 
G:=Group("(C2xC8):11D4");
 
// GroupNames label
 
G:=SmallGroup(128,1789);
 
// by ID
 
G=gap.SmallGroup(128,1789);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,521,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
 
// generators/relations
 

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