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## G = (C2×D4).5C8order 128 = 27

### 2nd non-split extension by C2×D4 of C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — (C2×D4).5C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C8○D4 — (C2×D4).5C8
 Lower central C1 — C22 — (C2×D4).5C8
 Upper central C1 — C2×C8 — (C2×D4).5C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — (C2×D4).5C8

Generators and relations for (C2×D4).5C8
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=ab-1, dcd-1=ab2c >

Subgroups: 172 in 114 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C2×C16 [×4], C2×C16 [×2], M5(2) [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4, C22⋊C16 [×4], C22×C16, C2×M5(2), C2×C8○D4, (C2×D4).5C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, D4○C16 [×2], (C2×D4).5C8

Smallest permutation representation of (C2×D4).5C8
On 64 points
Generators in S64
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 33)(30 34)(31 35)(32 36)
(1 13 9 5)(2 61 10 53)(3 15 11 7)(4 63 12 55)(6 49 14 57)(8 51 16 59)(17 33 25 41)(18 22 26 30)(19 35 27 43)(20 24 28 32)(21 37 29 45)(23 39 31 47)(34 38 42 46)(36 40 44 48)(50 62 58 54)(52 64 60 56)
(1 38)(2 27)(3 40)(4 29)(5 42)(6 31)(7 44)(8 17)(9 46)(10 19)(11 48)(12 21)(13 34)(14 23)(15 36)(16 25)(18 56)(20 58)(22 60)(24 62)(26 64)(28 50)(30 52)(32 54)(33 59)(35 61)(37 63)(39 49)(41 51)(43 53)(45 55)(47 57)
(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)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I 8J 8K 8L 8M 8N 8O 8P 16A ··· 16P 16Q ··· 16X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 8 8 8 8 16 ··· 16 16 ··· 16 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C8 C8 D4 M4(2) D4○C16 kernel (C2×D4).5C8 C22⋊C16 C22×C16 C2×M5(2) C2×C8○D4 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C2×C8 C2×C4 C2 # reps 1 4 1 1 1 6 2 12 4 4 4 16

Matrix representation of (C2×D4).5C8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 13 0 0 0 0 4 0 0 0 0 13
,
 0 13 0 0 4 0 0 0 0 0 0 4 0 0 13 0
,
 12 0 0 0 0 12 0 0 0 0 0 14 0 0 14 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,13,0,0,0,0,4,0,0,0,0,13],[0,4,0,0,13,0,0,0,0,0,0,13,0,0,4,0],[12,0,0,0,0,12,0,0,0,0,0,14,0,0,14,0] >;

(C2×D4).5C8 in GAP, Magma, Sage, TeX

(C_2\times D_4)._5C_8
% in TeX

G:=Group("(C2xD4).5C8");
// GroupNames label

G:=SmallGroup(128,845);
// by ID

G=gap.SmallGroup(128,845);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,102,124]);
// Polycyclic

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

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