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## G = (C2×C4).23D8order 128 = 27

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

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

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, C422C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.Q8, C4⋊D8, Q8.D4, C87D4, 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 ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q8 D8 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel (C2×C4).23D8 C22.4Q16 C23.65C23 C24.3C22 C2×D4⋊C4 C2×C4⋊C8 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×D4 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 1 2 1 1 2 2 2 2 4 6 4 1 1

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

 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 15 0 0 0 0 0 1 0 0 0 0 0 0 14 14 0 0 0 0 3 14 0 0 0 0 0 0 16 8 0 0 0 0 4 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 11 0 0 0 0 3 0
,
 1 0 0 0 0 0 16 16 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 0 0 0 0 0 11 0 0 0 0 3 0

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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