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G = (C2×C8).195D4order 128 = 27

163rd non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).195D4, (C2×C4)⋊2M4(2), C24.55(C2×C4), C4.115C22≀C2, (C22×C4).47Q8, C2.16(C89D4), C2.12(C86D4), C23.28(C4⋊C4), (C22×C4).281D4, C22.143(C4×D4), C22.51(C8○D4), C4.112(C22⋊Q8), (C22×C8).26C22, C2.C42.20C4, C23.312(C22×C4), (C2×C42).262C22, (C23×C4).247C22, C22.65(C2×M4(2)), C2.7(C23.8Q8), C2.10(C4⋊M4(2)), C22.7C4239C2, (C22×C4).1628C23, (C22×M4(2)).19C2, C4.132(C22.D4), C2.13(C42.6C22), (C2×C4⋊C8)⋊39C2, (C2×C4).50(C4⋊C4), (C2×C4).342(C2×Q8), (C2×C4).1526(C2×D4), (C4×C22⋊C4).14C2, (C2×C22⋊C8).21C2, (C2×C22⋊C4).38C4, C22.108(C2×C4⋊C4), (C2×C4).934(C4○D4), (C22×C4).117(C2×C4), SmallGroup(128,583)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — (C2×C8).195D4
C1C2C4C2×C4C22×C4C23×C4C22×M4(2) — (C2×C8).195D4
C1C23 — (C2×C8).195D4
C1C22×C4 — (C2×C8).195D4
C1C2C2C22×C4 — (C2×C8).195D4

Generators and relations for (C2×C8).195D4
 G = < a,b,c,d | a2=b8=1, c4=b4, d2=b2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=ab5, dcd-1=b6c3 >

Subgroups: 276 in 162 conjugacy classes, 68 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×6], C2×C4 [×2], C2×C4 [×10], C2×C4 [×20], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×6], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C22⋊C8, C2×C4⋊C8 [×2], C22×M4(2), (C2×C8).195D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C8○D4 [×2], C23.8Q8, C4⋊M4(2), C42.6C22, C89D4 [×2], C86D4 [×2], (C2×C8).195D4

Smallest permutation representation of (C2×C8).195D4
On 64 points
Generators in S64
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(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 56 11 19 5 52 15 23)(2 33 12 64 6 37 16 60)(3 50 13 21 7 54 9 17)(4 35 14 58 8 39 10 62)(18 42 51 26 22 46 55 30)(20 44 53 28 24 48 49 32)(25 57 45 38 29 61 41 34)(27 59 47 40 31 63 43 36)
(1 17 3 19 5 21 7 23)(2 58 4 60 6 62 8 64)(9 56 11 50 13 52 15 54)(10 37 12 39 14 33 16 35)(18 46 20 48 22 42 24 44)(25 36 27 38 29 40 31 34)(26 49 28 51 30 53 32 55)(41 63 43 57 45 59 47 61)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4R8A···8P
order12···2224···44···48···8
size11···1441···14···44···4

44 irreducible representations

dim11111111222222
type++++++++-
imageC1C2C2C2C2C2C4C4D4D4Q8M4(2)C4○D4C8○D4
kernel(C2×C8).195D4C22.7C42C4×C22⋊C4C2×C22⋊C8C2×C4⋊C8C22×M4(2)C2.C42C2×C22⋊C4C2×C8C22×C4C22×C4C2×C4C2×C4C22
# reps12112144422848

Matrix representation of (C2×C8).195D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
100000
0160000
0001600
004000
000012
00001016
,
010000
1600000
0001600
0013000
0000139
0000114
,
010000
100000
0001600
004000
0000139
0000104

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],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,16,0,0,0,0,0,0,0,1,10,0,0,0,0,2,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,16,0,0,0,0,0,0,0,13,11,0,0,0,0,9,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,16,0,0,0,0,0,0,0,13,10,0,0,0,0,9,4] >;

(C2×C8).195D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{195}D_4
% in TeX

G:=Group("(C2xC8).195D4");
// GroupNames label

G:=SmallGroup(128,583);
// by ID

G=gap.SmallGroup(128,583);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=b^2,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*c*d^-1=b^6*c^3>;
// generators/relations

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