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Elements of the group are displayed as permutations of degree 20.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $A_4^2:C_2^4.S_4$ | 1A | $1$ | $1$ | $A_4^2:C_2^4.S_4$ | 1A | 1A | $()$ |
| $A_4^2:C_2^4.S_4$ | 2A | $2$ | $1$ | $A_4^2:C_2^4.S_4$ | 1A | 2A | $(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2B | $2$ | $3$ | $A_4^2.C_2^5.C_2^2$ | 1A | 2B | $(13,15)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2C | $2$ | $3$ | $A_4^2.C_2^5.C_2^2$ | 1A | 2C | $(11,12)(13,15)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2D | $2$ | $3$ | $A_4^2.C_2^5.C_2^2$ | 1A | 2D | $(11,17)(12,18)(13,19)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2E | $2$ | $3$ | $A_4^2.C_2^5.C_2^2$ | 1A | 2E | $(11,18)(12,17)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2F | $2$ | $3$ | $A_4^2.C_2^5.C_2^2$ | 1A | 2F | $(9,10)(11,17)(12,18)(13,19)(14,16)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2G | $2$ | $3$ | $A_4^2.C_2^5.C_2^2$ | 1A | 2G | $(9,10)(11,18)(12,17)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2H | $2$ | $6$ | $A_4.C_2^5.C_6.C_2^2$ | 1A | 2H | $(2,7)(3,8)$ |
| $A_4^2:C_2^4.S_4$ | 2I | $2$ | $6$ | $C_2^3\times A_4^2.C_2^3$ | 1A | 2I | $(11,17)(12,18)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2J | $2$ | $6$ | $C_2^3\times A_4^2.C_2^3$ | 1A | 2J | $(9,10)(11,17)(12,18)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2K | $2$ | $6$ | $A_4.C_2^5.C_6.C_2^2$ | 1A | 2K | $(2,3)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2L | $2$ | $9$ | $C_2^4:C_3.C_2^4.C_2^3$ | 1A | 2L | $(1,5)(2,7)(3,8)(4,6)$ |
| $A_4^2:C_2^4.S_4$ | 2M | $2$ | $9$ | $C_2^4:C_3.C_2^4.C_2^3$ | 1A | 2M | $(1,4)(2,8)(3,7)(5,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2N | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2N | $(2,3)(7,8)(13,15)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2O | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2O | $(2,3)(7,8)(11,12)(13,15)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2P | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2P | $(2,3)(7,8)(11,17)(12,18)(13,19)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2Q | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2Q | $(2,3)(7,8)(11,17)(12,18)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2R | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2R | $(2,3)(7,8)(11,18)(12,17)(13,19)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2S | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2S | $(2,3)(7,8)(11,18)(12,17)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2T | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2T | $(2,3)(7,8)(9,10)(11,17)(12,18)(13,19)(14,16)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2U | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2U | $(2,3)(7,8)(9,10)(11,17)(12,18)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2V | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2V | $(2,3)(7,8)(9,10)(11,18)(12,17)(13,19)(14,16)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2W | $2$ | $18$ | $A_4.C_2^5.C_2^3$ | 1A | 2W | $(2,3)(7,8)(9,10)(11,18)(12,17)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2X | $2$ | $27$ | $C_2^6.C_2^5$ | 1A | 2X | $(1,4)(2,3)(5,6)(7,8)(11,12)(13,15)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2Y | $2$ | $27$ | $C_2^6.C_2^5$ | 1A | 2Y | $(1,4)(2,3)(5,6)(7,8)(9,10)(11,17)(12,18)(13,19)(14,16)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2Z | $2$ | $27$ | $C_2^6.C_2^5$ | 1A | 2Z | $(1,4)(2,3)(5,6)(7,8)(9,10)(11,18)(12,17)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AA | $2$ | $27$ | $C_2^6.C_2^5$ | 1A | 2AA | $(1,6)(2,8)(3,7)(4,5)(9,14)(10,16)(11,17)(12,18)$ |
| $A_4^2:C_2^4.S_4$ | 2AB | $2$ | $27$ | $C_2^6.C_2^5$ | 1A | 2AB | $(1,6)(2,8)(3,7)(4,5)(9,10)(14,16)$ |
| $A_4^2:C_2^4.S_4$ | 2AC | $2$ | $27$ | $C_2^6.C_2^5$ | 1A | 2AC | $(1,6)(2,8)(3,7)(4,5)(9,14)(10,16)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AD | $2$ | $36$ | $C_2\times C_2^4.(C_6.D_4)$ | 1A | 2AD | $(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2AE | $2$ | $36$ | $C_2\times C_2^4.(C_6.D_4)$ | 1A | 2AE | $(4,6)(7,8)$ |
| $A_4^2:C_2^4.S_4$ | 2AF | $2$ | $54$ | $C_2^4.C_2^6$ | 1A | 2AF | $(1,4)(2,3)(5,6)(7,8)(11,17)(12,18)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AG | $2$ | $54$ | $C_2^4.C_2^6$ | 1A | 2AG | $(1,4)(2,3)(5,6)(7,8)(9,10)(11,17)(12,18)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AH | $2$ | $108$ | $C_2^7:C_4$ | 1A | 2AH | $(5,6)(7,8)(13,15)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2AI | $2$ | $108$ | $C_2^7:C_4$ | 1A | 2AI | $(5,6)(7,8)(11,12)(13,15)(17,18)(19,20)$ |
| $A_4^2:C_2^4.S_4$ | 2AJ | $2$ | $108$ | $C_2^7:C_4$ | 1A | 2AJ | $(5,6)(7,8)(11,17)(12,18)(13,19)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2AK | $2$ | $108$ | $C_2^7:C_4$ | 1A | 2AK | $(5,6)(7,8)(11,18)(12,17)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AL | $2$ | $108$ | $C_2^7:C_4$ | 1A | 2AL | $(5,6)(7,8)(9,10)(11,17)(12,18)(13,19)(14,16)(15,20)$ |
| $A_4^2:C_2^4.S_4$ | 2AM | $2$ | $108$ | $C_2^7:C_4$ | 1A | 2AM | $(5,6)(7,8)(9,10)(11,18)(12,17)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AN | $2$ | $216$ | not computed | 1A | 2AN | $(5,6)(7,8)(11,17)(12,18)(13,20)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 2AO | $2$ | $216$ | not computed | 1A | 2AO | $(5,6)(7,8)(9,10)(11,17)(12,18)(13,20)(14,16)(15,19)$ |
| $A_4^2:C_2^4.S_4$ | 3A | $3$ | $16$ | $C_2^6.C_3^3.C_2$ | 3A | 1A | $(4,6,5)$ |
| $A_4^2:C_2^4.S_4$ | 3B | $3$ | $32$ | $C_6\times \PSOPlus(4,3)$ | 3B | 1A | $(9,13,12)(10,15,11)(14,19,17)(16,20,18)$ |
| $A_4^2:C_2^4.S_4$ | 3C | $3$ | $32$ | $(C_2\times C_6^2).S_4$ | 3C | 1A | $(3,8,7)(4,6,5)$ |
| $A_4^2:C_2^4.S_4$ | 3D | $3$ | $32$ | $(C_2\times C_6^2).S_4$ | 3D | 1A | $(3,8,7)(4,5,6)$ |
| $A_4^2:C_2^4.S_4$ | 3E1 | $3$ | $256$ | not computed | 3E-1 | 1A | $(3,8,7)(9,19,11)(10,20,12)(13,18,14)(15,17,16)$ |
| $A_4^2:C_2^4.S_4$ | 3E-1 | $3$ | $256$ | not computed | 3E1 | 1A | $(3,7,8)(9,11,19)(10,12,20)(13,14,18)(15,16,17)$ |
| $A_4^2:C_2^4.S_4$ | 3F | $3$ | $1024$ | not computed | 3F | 1A | $(3,7,8)(4,6,5)(9,18,19)(10,17,20)(11,13,14)(12,15,16)$ |
| $A_4^2:C_2^4.S_4$ | 3G | $3$ | $1024$ | not computed | 3G | 1A | $(1,4,6)(2,3,8)(9,13,12)(10,15,11)(14,19,17)(16,20,18)$ |