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G = (C2xC8):11D4  order 128 = 27

7th semidirect product of C2xC8 and D4 acting via D4/C2=C22

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

Aliases: (C2xC8):11D4, C8:D4:3C2, C8:8D4:43C2, C8.119(C2xD4), (C2xD4).216D4, C4:C4.26C23, (C2xQ8).171D4, C23.75(C2xD4), C2.D8:70C22, C4.Q8:56C22, C22:Q8:3C22, C4.69(C4:D4), (C2xC8).253C23, (C2xC4).261C24, (C22xC8):41C22, (C22xSD16):2C2, (C2xD4).64C23, C4.155(C22xD4), (C2xQ8).52C23, C2.16(D4oSD16), Q8:C4:53C22, C4:D4.19C22, C23.25D4:27C2, C23.38D4:34C2, C23.37D4:34C2, C22.86(C4:D4), (C2xM4(2)):53C22, (C22xC4).983C23, C22.521(C22xD4), D4:C4.129C22, C22.29C24.11C2, (C2xSD16).134C22, (C22xD4).349C22, (C22xQ8).282C22, C42:C2.110C22, C23.38C23:10C2, (C2xC8oD4):1C2, C4.28(C2xC4oD4), (C2xC4).129(C2xD4), C2.79(C2xC4:D4), (C2xC4).282(C4oD4), (C2xC4oD4).301C22, SmallGroup(128,1789)

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

Derived series
C1 — C2xC4 — (C2xC8):11D4
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C2xC4oD4 — C2xC8oD4 — (C2xC8):11D4
Lower central
C1 — C2 — C2xC4 — (C2xC8):11D4
Upper central
C1 — C22 — C2xC4oD4 — (C2xC8):11D4
Jennings
C1 — C2 — C2 — C2xC4 — (C2xC8):11D4

Generators and relations for (C2xC8):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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), SD16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C24, D4:C4, Q8:C4, C4.Q8, C2.D8, C42:C2, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C4:1D4, C4:Q8, C22xC8, C22xC8, C2xM4(2), C2xM4(2), C8oD4, C2xSD16, C2xSD16, C22xD4, C22xQ8, C2xC4oD4, C23.37D4, C23.38D4, C23.25D4, C8:8D4, C8:D4, C22.29C24, C23.38C23, C2xC8oD4, C22xSD16, (C2xC8):11D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, C2xC4:D4, D4oSD16, (C2xC8):11D4

Smallest permutation representation of (C2xC8):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+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4oD4D4oSD16
kernel(C2xC8):11D4C23.37D4C23.38D4C23.25D4C8:8D4C8:D4C22.29C24C23.38C23C2xC8oD4C22xSD16C2xC8C2xD4C2xQ8C2xC4C2
# reps111144111143144

Matrix representation of (C2xC8):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] >;
 

(C2xC8):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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