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G = (C2×C4).24D8order 128 = 27

17th non-split extension by C2×C4 of D8 acting via D8/C4=C22

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

Aliases: (C2×C4).24D8, C429C46C2, (C2×D4).117D4, (C2×C4).39SD16, C22.88(C2×D8), C2.16(C4⋊D8), C2.7(C4.4D8), (C22×C4).317D4, C23.926(C2×D4), C22.4Q1627C2, (C22×C8).84C22, C4.37(C422C2), C2.16(D4.D4), (C2×C42).380C22, C22.103(C2×SD16), C2.7(C22.D8), (C22×D4).92C22, C22.247(C4⋊D4), C22.148(C8⋊C22), C22.7C4213C2, (C22×C4).1460C23, C4.26(C22.D4), C2.6(C23.11D4), C22.93(C4.4D4), C2.7(C23.46D4), C22.137(C8.C22), C24.3C22.19C2, C2.9(C42.28C22), C22.116(C22.D4), (C2×C4).1059(C2×D4), (C2×D4⋊C4).17C2, (C2×C4).884(C4○D4), (C2×C4⋊C4).145C22, SmallGroup(128,803)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4).24D8
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×D4⋊C4 — (C2×C4).24D8
C1C2C22×C4 — (C2×C4).24D8
C1C23C2×C42 — (C2×C4).24D8
C1C2C2C22×C4 — (C2×C4).24D8

Generators and relations for (C2×C4).24D8
 G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=ac-1 >

Subgroups: 344 in 137 conjugacy classes, 50 normal (28 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×D4, C22.7C42, C22.4Q16 [×2], C429C4, C24.3C22, C2×D4⋊C4 [×2], (C2×C4).24D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], SD16 [×2], C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.11D4, C4⋊D8, D4.D4, C22.D8, C23.46D4, C4.4D8, C42.28C22, (C2×C4).24D8

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim1111112222244
type++++++++++-
imageC1C2C2C2C2C2D4D4D8SD16C4○D4C8⋊C22C8.C22
kernel(C2×C4).24D8C22.7C42C22.4Q16C429C4C24.3C22C2×D4⋊C4C22×C4C2×D4C2×C4C2×C4C2×C4C22C22
# reps11211222441011

Matrix representation of (C2×C4).24D8 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
010000
1600000
001000
000100
000048
00001313
,
1250000
12120000
0031400
003300
000012
0000016
,
100000
0160000
001000
0001600
000010
00001616

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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,1,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

(C2×C4).24D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{24}D_8
% in TeX

G:=Group("(C2xC4).24D8");
// GroupNames label

G:=SmallGroup(128,803);
// by ID

G=gap.SmallGroup(128,803);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,2028,1027,124]);
// Polycyclic

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

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