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G = C4⋊C4.7C8order 128 = 27

4th non-split extension by C4⋊C4 of C8 acting via C8/C4=C2

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

Aliases: C4⋊C4.7C8, C4⋊C1614C2, C4.19(C4⋊C8), C8.38(C4⋊C4), (C2×C8).62Q8, C8.42(C2×Q8), C8.135(C2×D4), (C2×C8).392D4, C22⋊C4.3C8, C8⋊C4.17C4, C2.5(D4○C16), (C22×C16).9C2, C23.21(C2×C8), C22.13(C4⋊C8), (C2×C16).65C22, C42.169(C2×C4), (C2×C8).628C23, (C4×C8).321C22, (C2×C4).51M4(2), C4.66(C2×M4(2)), C42⋊C2.21C4, (C2×M5(2)).24C2, (C2×M4(2)).33C4, C22.49(C22×C8), C82M4(2).20C2, (C22×C8).579C22, C4.80(C2×C4⋊C4), C2.14(C2×C4⋊C8), (C2×C4).26(C2×C8), (C2×C8).195(C2×C4), (C2×C4).138(C4⋊C4), (C2×C4).613(C22×C4), (C22×C4).409(C2×C4), SmallGroup(128,883)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊C4.7C8
C1C2C4C8C2×C8C22×C8C82M4(2) — C4⋊C4.7C8
C1C22 — C4⋊C4.7C8
C1C2×C8 — C4⋊C4.7C8
C1C2C2C2C2C4C4C2×C8 — C4⋊C4.7C8

Generators and relations for C4⋊C4.7C8
 G = < a,b,c | a4=b4=1, c8=a2, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 100 in 80 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], C23, C16 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×4], C2×C16 [×2], M5(2) [×2], C42⋊C2, C22×C8, C2×M4(2), C4⋊C16 [×4], C82M4(2), C22×C16, C2×M5(2), C4⋊C4.7C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, D4○C16 [×2], C4⋊C4.7C8

Smallest permutation representation of C4⋊C4.7C8
On 64 points
Generators in S64
(1 46 9 38)(2 47 10 39)(3 48 11 40)(4 33 12 41)(5 34 13 42)(6 35 14 43)(7 36 15 44)(8 37 16 45)(17 53 25 61)(18 54 26 62)(19 55 27 63)(20 56 28 64)(21 57 29 49)(22 58 30 50)(23 59 31 51)(24 60 32 52)
(1 19 42 59)(2 52 43 28)(3 21 44 61)(4 54 45 30)(5 23 46 63)(6 56 47 32)(7 25 48 49)(8 58 33 18)(9 27 34 51)(10 60 35 20)(11 29 36 53)(12 62 37 22)(13 31 38 55)(14 64 39 24)(15 17 40 57)(16 50 41 26)
(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)| (1,46,9,38)(2,47,10,39)(3,48,11,40)(4,33,12,41)(5,34,13,42)(6,35,14,43)(7,36,15,44)(8,37,16,45)(17,53,25,61)(18,54,26,62)(19,55,27,63)(20,56,28,64)(21,57,29,49)(22,58,30,50)(23,59,31,51)(24,60,32,52), (1,19,42,59)(2,52,43,28)(3,21,44,61)(4,54,45,30)(5,23,46,63)(6,56,47,32)(7,25,48,49)(8,58,33,18)(9,27,34,51)(10,60,35,20)(11,29,36,53)(12,62,37,22)(13,31,38,55)(14,64,39,24)(15,17,40,57)(16,50,41,26), (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( (1,46,9,38)(2,47,10,39)(3,48,11,40)(4,33,12,41)(5,34,13,42)(6,35,14,43)(7,36,15,44)(8,37,16,45)(17,53,25,61)(18,54,26,62)(19,55,27,63)(20,56,28,64)(21,57,29,49)(22,58,30,50)(23,59,31,51)(24,60,32,52), (1,19,42,59)(2,52,43,28)(3,21,44,61)(4,54,45,30)(5,23,46,63)(6,56,47,32)(7,25,48,49)(8,58,33,18)(9,27,34,51)(10,60,35,20)(11,29,36,53)(12,62,37,22)(13,31,38,55)(14,64,39,24)(15,17,40,57)(16,50,41,26), (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([(1,46,9,38),(2,47,10,39),(3,48,11,40),(4,33,12,41),(5,34,13,42),(6,35,14,43),(7,36,15,44),(8,37,16,45),(17,53,25,61),(18,54,26,62),(19,55,27,63),(20,56,28,64),(21,57,29,49),(22,58,30,50),(23,59,31,51),(24,60,32,52)], [(1,19,42,59),(2,52,43,28),(3,21,44,61),(4,54,45,30),(5,23,46,63),(6,56,47,32),(7,25,48,49),(8,58,33,18),(9,27,34,51),(10,60,35,20),(11,29,36,53),(12,62,37,22),(13,31,38,55),(14,64,39,24),(15,17,40,57),(16,50,41,26)], [(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 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12222244444444448···88888888816···1616···16
size11112211112244441···1222244442···24···4

56 irreducible representations

dim11111111112222
type++++++-
imageC1C2C2C2C2C4C4C4C8C8D4Q8M4(2)D4○C16
kernelC4⋊C4.7C8C4⋊C16C82M4(2)C22×C16C2×M5(2)C8⋊C4C42⋊C2C2×M4(2)C22⋊C4C4⋊C4C2×C8C2×C8C2×C4C2
# reps141114228822416

Matrix representation of C4⋊C4.7C8 in GL4(𝔽17) generated by

16000
01600
0040
001613
,
161600
2100
0018
00416
,
161600
0100
00120
00012
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,16,0,0,0,13],[16,2,0,0,16,1,0,0,0,0,1,4,0,0,8,16],[16,0,0,0,16,1,0,0,0,0,12,0,0,0,0,12] >;

C4⋊C4.7C8 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._7C_8
% in TeX

G:=Group("C4:C4.7C8");
// GroupNames label

G:=SmallGroup(128,883);
// by ID

G=gap.SmallGroup(128,883);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,723,102,124]);
// Polycyclic

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

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