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G = (C2×C8)⋊12D4  order 128 = 27

8th semidirect product of C2×C8 and D4 acting via D4/C2=C22

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

Aliases: (C2×C8)⋊12D4, C8⋊2D4⋊3C2, C8⋊7D4⋊34C2, C8.109(C2×D4), (C2×D4).217D4, C2.10(D4○D8), (C22×D8)⋊16C2, C4⋊D4⋊3C22, C4⋊C4.27C23, (C2×Q8).172D4, C23.76(C2×D4), C2.D8⋊47C22, C4.Q8⋊51C22, C4.70(C4⋊D4), (C2×C8).254C23, (C2×C4).262C24, (C22×C8)⋊31C22, (C2×D4).65C23, C4.156(C22×D4), D4⋊C4⋊52C22, (C2×D8).119C22, C22.29C24⋊10C2, C23.37D4⋊35C2, C23.25D4⋊28C2, C22.87(C4⋊D4), (C2×M4(2))⋊54C22, (C22×C4).984C23, C22.522(C22×D4), (C22×D4).350C22, C42⋊C2.111C22, (C2×C8○D4)⋊2C2, C4.29(C2×C4○D4), (C2×C4).130(C2×D4), C2.80(C2×C4⋊D4), (C2×C4).283(C4○D4), (C2×C4○D4).302C22, SmallGroup(128,1790)

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

Derived series
C1 — C2×C4 — (C2×C8)⋊12D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊12D4
Lower central
C1 — C2 — C2×C4 — (C2×C8)⋊12D4
Upper central
C1 — C22 — C2×C4○D4 — (C2×C8)⋊12D4
Jennings
C1 — C2 — C2 — C2×C4 — (C2×C8)⋊12D4

Generators and relations for (C2×C8)⋊12D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 620 in 262 conjugacy classes, 100 normal (20 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, C2×C8, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×D8, C22×D4, C2×C4○D4, C23.37D4, C23.25D4, C8⋊7D4, C8⋊2D4, C22.29C24, C2×C8○D4, C22×D8, (C2×C8)⋊12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○D8, (C2×C8)⋊12D4

Smallest permutation representation of (C2×C8)⋊12D4
►On 32 points
Generators in S32
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E···8J
order122222222222444444444488888···8
size111122448888222244888822224···4

32 irreducible representations

dim1111111122224
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4C4○D4D4○D8
kernel(C2×C8)⋊12D4C23.37D4C23.25D4C8⋊7D4C8⋊2D4C22.29C24C2×C8○D4C22×D8C2×C8C2×D4C2×Q8C2×C4C2
# reps1214421143144

Matrix representation of (C2×C8)⋊12D4 ►in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
0130000
1300000
0031400
003300
0000314
000033
,
400000
0130000
00001414
0000143
003300
0031400
,
0130000
400000
003300
0031400
00001414
0000143

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,14,14,0,0,0,0,14,3,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;
 

(C2×C8)⋊12D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8)\rtimes_{12}D_4
 
% in TeX
 
G:=Group("(C2xC8):12D4");
 
// GroupNames label
 
G:=SmallGroup(128,1790);
 
// by ID
 
G=gap.SmallGroup(128,1790);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,521,2804,172]);
 
// 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^-1,d*c*d=c^-1>;
 
// generators/relations
 

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