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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

Aliases: (C2×D4).5C8, (C2×Q8).5C8, (C22×C16)⋊2C2, C8.132(C2×D4), (C2×C8).388D4, C23.9(C2×C8), C22⋊C1614C2, C2.4(D4○C16), (C2×M5(2))⋊7C2, C8.29(C22⋊C4), C4.14(C22⋊C8), (C2×C8).625C23, (C2×C16).61C22, C4.63(C2×M4(2)), (C2×C4).49M4(2), (C2×M4(2)).28C4, C22.4(C22⋊C8), C22.47(C22×C8), (C22×C8).414C22, (C2×C4).24(C2×C8), (C2×C8).193(C2×C4), (C2×C8○D4).16C2, (C2×C4○D4).17C4, C2.23(C2×C22⋊C8), C4.114(C2×C22⋊C4), (C22×C4).286(C2×C4), (C2×C4).610(C22×C4), (C2×C4).266(C22⋊C4), SmallGroup(128,845)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — (C2×D4).5C8
C1C2C4C8C2×C8C22×C8C2×C8○D4 — (C2×D4).5C8
C1C22 — (C2×D4).5C8
C1C2×C8 — (C2×D4).5C8
C1C2C2C2C2C4C4C2×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 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12222222444444448···88888888816···1616···16
size11112244111122441···1222244442···24···4

56 irreducible representations

dim111111111222
type++++++
imageC1C2C2C2C2C4C4C8C8D4M4(2)D4○C16
kernel(C2×D4).5C8C22⋊C16C22×C16C2×M5(2)C2×C8○D4C2×M4(2)C2×C4○D4C2×D4C2×Q8C2×C8C2×C4C2
# reps14111621244416

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

16000
01600
0010
0001
,
4000
01300
0040
00013
,
01300
4000
0004
00130
,
12000
01200
00014
00140
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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