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G = (C2×C4).23D8  order 128 = 27

16th non-split extension by C2×C4 of D8 acting via D8/C4=C22

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

Aliases: (C2×C4).23D8, (C2×C8).51D4, C4⋊C4.108D4, (C2×D4).13Q8, C22.87(C2×D8), C2.20(C8⋊D4), C2.15(C4⋊D8), C2.15(C8⋊7D4), C2.8(D4⋊Q8), (C22×C4).156D4, C23.924(C2×D4), C2.11(D4.Q8), C4.37(C22⋊Q8), C4.155(C4⋊D4), C22.4Q16⋊26C2, (C22×C8).82C22, C4.35(C42⋊2C2), C22.118(C4○D8), (C2×C42).378C22, C2.21(Q8.D4), (C22×D4).90C22, C2.8(C23.Q8), C22.245(C4⋊D4), C22.146(C8⋊C22), (C22×C4).1458C23, C23.65C23⋊8C2, C22.111(C22⋊Q8), C22.135(C8.C22), C24.3C22.18C2, (C2×C4⋊C8)⋊21C2, (C2×C2.D8)⋊9C2, (C2×C4).285(C2×Q8), (C2×C4).1057(C2×D4), (C2×D4⋊C4).16C2, (C2×C4).622(C4○D4), (C2×C4⋊C4).143C22, SmallGroup(128,799)

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

Derived series
C1 — C22×C4 — (C2×C4).23D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×D4⋊C4 — (C2×C4).23D8
Lower central
C1 — C2 — C22×C4 — (C2×C4).23D8
Upper central
C1 — C23 — C2×C42 — (C2×C4).23D8
Jennings
C1 — C2 — C2 — C22×C4 — (C2×C4).23D8

Generators and relations for (C2×C4).23D8
 G = < a,b,c,d | a2=b8=c4=1, d2=b6, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=b6c-1 >

Subgroups: 344 in 142 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×D4⋊C4, C2×C4⋊C8, C2×C2.D8, (C2×C4).23D8
Quotients: C1, C2, C22, D4, Q8, C23, D8, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C42⋊2C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.Q8, C4⋊D8, Q8.D4, C8⋊7D4, C8⋊D4, D4⋊Q8, D4.Q8, (C2×C4).23D8

Smallest permutation representation of (C2×C4).23D8
►On 64 points
Generators in S64
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 22 51 11)(2 21 52 10)(3 20 53 9)(4 19 54 16)(5 18 55 15)(6 17 56 14)(7 24 49 13)(8 23 50 12)(25 40 64 48)(26 39 57 47)(27 38 58 46)(28 37 59 45)(29 36 60 44)(30 35 61 43)(31 34 62 42)(32 33 63 41)

(1 9 7 15 5 13 3 11)(2 64 8 62 6 60 4 58)(10 46 16 44 14 42 12 48)(17 34 23 40 21 38 19 36)(18 55 24 53 22 51 20 49)(25 50 31 56 29 54 27 52)(26 39 32 37 30 35 28 33)(41 57 47 63 45 61 43 59)
 
G:=sub<Sym(64)| (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,22,51,11)(2,21,52,10)(3,20,53,9)(4,19,54,16)(5,18,55,15)(6,17,56,14)(7,24,49,13)(8,23,50,12)(25,40,64,48)(26,39,57,47)(27,38,58,46)(28,37,59,45)(29,36,60,44)(30,35,61,43)(31,34,62,42)(32,33,63,41), (1,9,7,15,5,13,3,11)(2,64,8,62,6,60,4,58)(10,46,16,44,14,42,12,48)(17,34,23,40,21,38,19,36)(18,55,24,53,22,51,20,49)(25,50,31,56,29,54,27,52)(26,39,32,37,30,35,28,33)(41,57,47,63,45,61,43,59)>;
 
G:=Group( (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,22,51,11)(2,21,52,10)(3,20,53,9)(4,19,54,16)(5,18,55,15)(6,17,56,14)(7,24,49,13)(8,23,50,12)(25,40,64,48)(26,39,57,47)(27,38,58,46)(28,37,59,45)(29,36,60,44)(30,35,61,43)(31,34,62,42)(32,33,63,41), (1,9,7,15,5,13,3,11)(2,64,8,62,6,60,4,58)(10,46,16,44,14,42,12,48)(17,34,23,40,21,38,19,36)(18,55,24,53,22,51,20,49)(25,50,31,56,29,54,27,52)(26,39,32,37,30,35,28,33)(41,57,47,63,45,61,43,59) );
 
G=PermutationGroup([[(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,22,51,11),(2,21,52,10),(3,20,53,9),(4,19,54,16),(5,18,55,15),(6,17,56,14),(7,24,49,13),(8,23,50,12),(25,40,64,48),(26,39,57,47),(27,38,58,46),(28,37,59,45),(29,36,60,44),(30,35,61,43),(31,34,62,42),(32,33,63,41)], [(1,9,7,15,5,13,3,11),(2,64,8,62,6,60,4,58),(10,46,16,44,14,42,12,48),(17,34,23,40,21,38,19,36),(18,55,24,53,22,51,20,49),(25,50,31,56,29,54,27,52),(26,39,32,37,30,35,28,33),(41,57,47,63,45,61,43,59)]])
 

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim1111111222222244
type++++++++++-++-
imageC1C2C2C2C2C2C2D4D4D4Q8D8C4○D4C4○D8C8⋊C22C8.C22
kernel(C2×C4).23D8C22.4Q16C23.65C23C24.3C22C2×D4⋊C4C2×C4⋊C8C2×C2.D8C4⋊C4C2×C8C22×C4C2×D4C2×C4C2×C4C22C22C22
# reps1111211222246411

Matrix representation of (C2×C4).23D8 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
16150000
010000
00141400
0031400
0000168
000041
,
100000
010000
005500
0051200
0000011
000030
,
100000
16160000
005500
0012500
0000011
000030

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,16,4,0,0,0,0,8,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,0,3,0,0,0,0,11,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,0,3,0,0,0,0,11,0] >;
 

(C2×C4).23D8 in GAP, Magma, Sage, TeX 

(C_2\times C_4)._{23}D_8
 
% in TeX
 
G:=Group("(C2xC4).23D8");
 
// GroupNames label
 
G:=SmallGroup(128,799);
 
// by ID
 
G=gap.SmallGroup(128,799);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,512,422,387,2019,521,248,2804,718,172,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^6,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=b^6*c^-1>;
 
// generators/relations
 

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