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## G = (C2×C4)⋊6D8order 128 = 27

### 1st semidirect product of C2×C4 and D8 acting via D8/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C4)⋊6D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — (C2×C4)⋊6D8
 Lower central C1 — C2 — C2×C4 — (C2×C4)⋊6D8
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊6D8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊6D8

Generators and relations for (C2×C4)⋊6D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=ab-1, dcd=c-1 >

Subgroups: 484 in 192 conjugacy classes, 68 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×10], D4 [×12], C23, C23 [×16], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], D8 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C24 [×2], C4×C8 [×2], D4⋊C4 [×4], C2.D8 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C22×D4 [×2], C24.3C22 [×2], C2×C4×C8, C2×D4⋊C4 [×2], C2×C2.D8, C22×D8, (C2×C4)⋊6D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], D8 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×D8 [×2], C4○D8 [×2], C24.3C22, C4×D8 [×2], C87D4 [×2], C84D4, C8.12D4, (C2×C4)⋊6D8

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M 4N 4O 4P 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 8 8 2 ··· 2 8 8 8 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D8 C4○D4 C4○D8 kernel (C2×C4)⋊6D8 C24.3C22 C2×C4×C8 C2×D4⋊C4 C2×C2.D8 C22×D8 C2×D8 C2×C8 C22×C4 C2×C4 C2×C4 C22 # reps 1 2 1 2 1 1 8 6 2 8 4 8

Matrix representation of (C2×C4)⋊6D8 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 4 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 13
,
 16 0 0 0 0 0 14 3 0 0 0 14 14 0 0 0 0 0 14 3 0 0 0 14 14
,
 1 0 0 0 0 0 14 14 0 0 0 14 3 0 0 0 0 0 3 3 0 0 0 3 14

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,13],[16,0,0,0,0,0,14,14,0,0,0,3,14,0,0,0,0,0,14,14,0,0,0,3,14],[1,0,0,0,0,0,14,14,0,0,0,14,3,0,0,0,0,0,3,3,0,0,0,3,14] >;

(C2×C4)⋊6D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_6D_8
% in TeX

G:=Group("(C2xC4):6D8");
// GroupNames label

G:=SmallGroup(128,702);
// by ID

G=gap.SmallGroup(128,702);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,436,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations

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